Answer: X = 8 or x = 4 ------------- I did this problem as follows: X 8 - + - = 3 4. X (LCD = 4X) X^2 + 32 3 ------ = - 4X 1 Cross multiplying, X^2 + 32 = 12X X^2 -12X + 32 = 0 Factoring, (X-4)(X-8) = 0 Either X-4 = 0 OR X-8 = 0 If X-4 = 0, X = 4 If X-8 = 0, X = 8 Answer: X = 4 or X = 8
I think I've solved so many of these this lifetime I could see the answers in front of me. The first obvious one was 4. Because it was a quadratic equation and I instinctively knew the answers were whole numbers figuring out 8 took another 35 seconds (I was timing). Not exactly procedure I'll admit.
@@justme2580 I did this problem as follows: X 8 - + - = 3 4. X (LCD = 4X) X^2 + 32 3 ------ = - 4X 1 Cross multiplying, X^2 + 32 = 12X X^2 -12X + 32 = 0 Factoring, (X-4)(X-8) = 0 Either X-4 = 0 OR X-8 = 0 If X-4 = 0, X = 4 If X-8 = 0, X = 8 Answer: X = 4 or X = 8
To add fractions you need a common denominator like you were taught in 7th grade. To do this make the common denominator 4 * x. Then multiply each term by one to get (x/4 * x/x) + (8/x * 4/4) = 3 to get the equation (x^2 + 32) / 4x = 3 . Then multiply both sides by 4x to get x^2 + 32 = 12x and put equation in standard form of x^2 - 12x +32 = 0 . Then factor to (x - 4)*(x - 8) = 0 to get the answer of 4, 8. There is no need to find the lowest common denominator or to multiply the whole equation my some magic LCD of 4*x*1.
Yes it is x=4 and x=8 both work, took me 2 minutes in my head. Step 1 common denominator (4x) x^2/4x +8(4)/4x = 3. Step 2 multiply both sides by 4x, x^2 + 32 = 3(4x) then x^2 + 32 = 12x. Step 3 subtract 12x from both sides x^2 - 12x + 32 = 0. Step 4 factor (x - 4)(x - 8) = 0. Step 5 solve for x in each set x - 4 =0, x - 8 =0. Step 6a add 4 to each side in x - 4 = 0 x - 4 + 4 = 0 + 4, x=4 Step 6d add 8 to both sides in x - 8 = 0, x - 8 + 8 = 0 + 8 x=8 Step 7 test results by substituting results in origional equation. A. 4/4 +8/4 = 3 1 + 2 = 3, 3 = 3,B. 8/4 + 8/8 = 3, 2 + 1 = 3, 3 = 3. Conclusion both 8 and 4 are solutions for x.
The answer is x=4 and/or x=8. It's simple by doing it in steps. The first step is subtract 8/x from each side of the equation, so x/4 = 3 - 8/x The next step is to multiply both sides by 4 so x = 4(3 - 8/x) or 12 - 32/x Then multiply each side by a factor of x. X^2 = 12x - 32 Then isolate the equation to one side. X^2-12x+32=0 This factors out to (x - 4)(x-8)=0 or x=4 and/or x=8
@davidseed2939 I remember in school the instructor wanted students to show their work to explain how they arrived at the answer, so the solution was just as important if not moreso than the correct answer(s).
I'm 5 years younger and when I was in school, the minute girls hit teen years, math was for boys, not girls for the most part. If you liked math you were not popular.
I try and try to listen to your video but you go off on a tangent too much. Six minutes into the video and you have not begun solving yet. Would love to learn from you but you just talk and talk and talk and ...................................
@@lamper2 I do, but sometimes he will say something pertinent to me but than he goes off again. Not a problem, just won't watch any of this videos anymore. Anywho I'm retired living comfortably, but I love learning.
Same 😊 I was always in advanced math in school and I quickly realized that "I still have it". I also realized that I still have the same problem. I was able to plug in a number by eyeballing it and the number was right but I would not be able to show my work. And I always got deductions for not properly showing my work 😅
I'm using my UA-cam Premium subscription to learn freestyle swimming, math, Hungarian, Greek, Mandarin Chinese, and other things. But being 72 years old I want each episode to be as succinct as possible without unnecessary talk
He’s TEACHING slow enough for children & teenagers to grasp. I think many of you look at this to test yourself & memory which is not what it’s intended for. Unfortunately, many folks being paid as teachers in public school are not teaching to educate the kids or help them comprehend the information. I feel he has a real grasp on teaching.
Followed quite a few of his but i give up screaming here. Too much nervous energy and chat, and oh by the way …..not easy to take notes when he’s going off on digressions and sales pitches in between youtube ads. Too frenetic and nervous chaos.🤯 I’m sure a ton of content you have, but kindly remember kiss. Stop telling us about that and get on with showing us instead of interrupting yourself every 5seconds. I’m going back to the books after this.
I don't even know why... but, I found this challenge so satisfying... I actually remembered... I'm 1966. Thank you for the confidence boost. Makes me feel like I can help my grandkids.
x/4 + 8/x = 3 -> x^2/4x + 32/4x - 3 = 0 -> ((x/2+32)/4x) - 3 = 0 -> x^2-12x+32 (Quadratic equation) Given this quadratic equation we can solve for solutions to x using the quadratic formula: (-b+-sqrt(d))/2a where d=b^2-4ac is the discriminant to the quadratic equation: x = (-b+-sqrt(b^2-4ac))/2a = (-(-12)+-sqrt((-12)^2-4*1*32))/2*1 = (12+-sqrt(16))/2 = 6+-2 The discriminant (16) is greater than 0 so we get 2 solutions: x1=6+2=8 x2=6-2=4 Intuitively the solution could also have been found by simply looking at the equation x/4 + 8/x = 3 and realizing that probably the solutions are 4 and 8 by substituting into x and doing a bit of simple math: 4/4+8/4=3 as well as 8/4+8/8=3. With more complex equations and/or unintuitive values of x the best method is to use formula for the discriminant and the quadratic formula.
1)- Thank you for taking the time, but you took us around the world and brought us back to the beginning. 2) - Algebra is not as easy as you have tried to teach your pupils or students. 3) recommend you to teach your skills in an easy meeting!!!!.
Agreed! Factoring is trial and error guess-work - just hoping that known factors of 32 just happen to add up to a nice round number like 12. 40 years of working with computers tells me that a simple function works best: x = solve_quadratic(1,-12,32) - sorted! 🤓
To solve the equation x/4 + 8/x = 3, we can first multiply both sides of the equation by 4x to eliminate the fractions. (x/4 + 8/x) * 4x = 3 * 4x x^2 + 32 = 12x Next, rearrange the equation to set it equal to zero: x^2 - 12x + 32 = 0 Now, we can factor this quadratic equation: (x - 8)(x - 4) = 0 Setting each factor equal to zero gives us the possible solutions for x: x - 8 = 0 or x - 4 = 0 x = 8 or x = 4 Therefore, the solutions to the equation x/4 + 8/x = 3 are x = 8 or x = 4.
taught math to most of my engineering class for 4 years. Went to work and became a digital designer who uses almost no math. I am 77 and love relearning, thank you!
4 was the first number that made any sense in my mind and without having much of an understanding of algebra or the other methods used to solve this problem.
Guess and check is quickest in this case. I need 2 numbers that add up to 3. Hmmm 1+2=3. Looks like 4 works in that case. Algebra is not always needed, sometimes a little number sense can work.
I did that too, but stopped at 4. A little more persistence could have given "What about 8?", since 8/8 is 1, and 8/4 is 2, and 2+1=3. Once you see it explained, it seems too simple.
I am happy to inform you that your lessons have produced solid results. 6 months ago I would have given up on this one and now I can solve it using algebra. Now I'll watch the video to see if my method was the right one but I'm confident it was. :) Thanks!
@@marscience7819 Yeap! But if you want to deal with tougher equations you need to learn algebra. In fact, I did not even think about plugging in the number 4 when I saw this problem. In other words I went from thinking like you did- totally fine- to thinking algebra first. So, I stand corrected in the sense that I would not have given up on the problem but I don't think the same way since I worked on my algebra skills which are still a work in progess BTW. ;) Try this one and see how you are doing without algebra: x/89 + 143/x = 542 Just a hint: number 4 won't work. ;)
This was a good one. I labored for a long time before finally realizing that I had to get that x out of the denominator. So here comes the quadratic equation, and it was simple to solve.
@@marscience7819 Yeah, I kind of did a facepalm when I got the answers. Sure, one number plus another number equals 3. The first choice would be that one fraction is 1 and the other is 2. That would have been so easy. But I knew this was an algebra problem, so I tried to use algebra to solve it.
I really like your video. You are very thorough and try to point out the complexities by referring to many points of solving the equations. Yes it may seem tedious but it helps me. I still have some trouble knowing when I need to look for lowest common multiplier versus lowest common denominator. I think it takes a lot of patience and determination to get good at this stuff. I am a biologist and though I've had some training in math, I was always kind of slow, but usually I could get through the courses with a good grade. Now I am retired and I find it a fun game to redirect my attention to algebra and other math, so I hope you and I will keep enjoying the challenges. Keep up the good work, and best wishes!
No need for complicated algebra. Use cross reduction and you get 1+2=3. The X’s cancel each other out, 4 and 8 a divisible by 4 leaving you with 1 over 1 and 2 over 1.
@@thomasmaughan4798 so, I simply solved the equation. So, you’re right I didn’t solve for X. However, cross reduction is a form of simplifying fractions to their lowest form and does work for addition and subtraction of fractions.
@@thomasmaughan4798 no, I don’t build bridges. However, if I did, I wouldn’t be using algebra to build it. I would use exact numbers, and if a number wasn’t known, I would go out and measure it again myself.
I took algebra in my first year in high school as a freshman. First we find the common denominator, which is 4x. Then, we multiply the 4x both sides which gives us x^2 + 32 = 12x. Then we make the equation to equal 0 by subtracting 12x both sides, which is gives us a quadratic equation of x^2 - 12x + 32 = 0. Thirdly, we factor the trinomial into two binomials using the FOIL method which is (x-4)(x-8) = 0. Finally we make each equation to equal 0, which is x-4=0 and/or x-8 = 0 and then solve for x. We get x = 4, x = 8.
The key to solving is to get all of the terms under a common denominator. The result is a quadratic equation in the numerator, which is easily factored into the two solutions.
I had to retake an algebra course in my late thirties since it was a pre-requisite to a university certificate I was taking. I am now 67 and I solved the problem like you in seconds. He had me doubting myself since he taking so long to explain it.
Well, I did it by taking care of the fractions first and then used the cross product to solve the problem. At the end I came up with the right equation anyway i.e. x Squared + 32 = 12x Then I substracted 12x from both sides, factored and came up with (x -4) (x - 8) = 0 and voilà! But I could have done it by multiplying the whole equation by 4x as you did. When I see fractions and an (=) sign I always think cross product. ;)
This particular equation can also be factored into 2 Binomials sq. by each other (x-4)*(x-8)=0 and {x = 4; x = 8} x^2 - 12x + 32 = 0 x/4 * 4x/1 + 8/x * 4x/1 = 3 * 4x fours cancel out, and leaves you with x^2; 8/x * 4x/1... the x's cross-cancel and leaves you with 8 * 4 = 32; Then, whatever you do to one side always do it on the other side of the "=". that gives you 12x quadratic equation: Ax^2 + Bx + C = 0 (in standard form) or (-)b±√b^2 - 4*a*c/2*a
Super simple, I would expect a c student to get this right in less than 20 seconds. 1:) Bow tie method: (x² + 32) / 4x = 3 2:) get rid of fraction by multiplication: [(x² + 32) / 4x ] * 4x = 3 * 4x x² + 32 = 12x 3:) set equation equal to zero and solve via foil x² -12x + 32 = 0 (x-8)(x-4) = 0 x = 8 and x = 4. Like I said super simple if you know what you're doing.
This is always where I got in trouble in school. They always wanted to show the work. They wanted to see how we solved it. That takes too long. I just look at the problem and my brain figures out the answer. This happens almost instantly
Thank you for your channel. I always loved math and now use your channel for brain exercises (I am 74). Shoutout to all my math teachers. I went as far as calculus -
If X squared over 4X + 32 over 4X = 3, then cross multiplication would mean that X squared + 32 = 3(4X), simplified to X squared + 32 = 12X. Then X squared minus X + 32 = 0. It then becomes (X-4)(X-8)=0. X-4=0, then X = 4. X - 8 = 0, then X = 8.
It's simple enough, that I don't need to create the full quadratic equation. The mean of the roots is 3 (constant term) * 4 (recibrocal of linear term) / 2 = 12/2 = 6. Obviously x = 4 and x = 8 result in 1+2 or 2+1, both are 3. Or you calculate 8 (1/x term) * 4 (see above) = 32 = x_1 * x_2. Or you take the long way with x^2 - 12 x + 32 = 0 => x = -(-12)/2 +/- sqrt(36 - 32) = 6 +/- 2.
What you are seeing is the schtick. He is an elite mathematician and wants to give you a real math experience. The "prollem" looks simple and turns out it is rife with pitfalls. You, therefore, can't solve it and he steps in step by step and solves it. This is how you learn math. If you want to play childhood T-ball, this is not the guy. He's gonna test you and, frankly, I like that. He also designs tests for a living, so he's going to know how folks get sidetracked. That is valuable information! Love, love, love this dude!
Without watching the video, this one is quite easy. You are allowed to multiply both members by any value without changing the equation. Let's do it by x. Which gives us 1/4 x^2 -3x +8 =0. ∆=b^2-4ac = 9 - 8 = 1. x1&2 = (-b +- √∆)/2a = (3 +- √1)/1/2 = 8 & 4.
Multiply by 4x to eliminate denominators. You get a quadratic; apply the quadratic formula. Or, even better, in this case the quadratic easily factors into a product of linear factors (x-4)(x-8).
I think I knew this sinds 1962: solution x =4 and x=8; by writing first part as the sum with the common denominator 4x and multiply by 4x you get the quadratic equation:ax^2+bx+c= x^2-12x+32=0; D=b^2-4ac; x=(-b±√D)/2a (D is the Discriminant)
you did it the hard way with the -b part, but yes technically that is correct. Now, if the test giver had requested foil or factoring, then no this would be wrong.
The answer is simply x=4 and x=8 Simply find the common denominator if you have the denominator of 4x multiply to both numerator and do the process of solving equations.. So x squared + 32 =12x So get the value of x it should be x=4 and x=8
This problem is EXTREMELY SIMPLE!! The result MUST equal 3, and "X" must be the same number. In this case "X" = 4. So, 4/4 = 1 + 8/4 = 2 Guess what? 1 + 2 = 3 Kindergarten wasn't wasted on me.
Inspection shows x=4 is one solution, with the 2nd term larger than the 1st. Increasing x makes the 1st term larger than the 2nd, so x=8 is another solution. Check video. Eventually found both answers after 17 min. Jeez! 3,500 people actually liked this!
Because of algebra I know that x will be the same number and that there might be more than one choice. Basic knowledge of fractions told me that if I could get fractions equal to 1 and 2 it would work because 1+2=3 and 2+1=3. 4/4=1 and 8/4=2 so one solution is 4. 4/8=2 and 8/8=1 Therefore the second solution is 8.
I multipled both sides of the equation by x to get the variable out of the denominator. Then you get a quadratic equation. I moved everything to one side of the equation so it’s equal to 0. Then I factored to get (x-8)(x-4)=0. So the solutions are 8 and 4.
When solving equations you move terms from one side of the equals sign to the other and change the sign. Positive terms become negative, division becomes multiplication etc it is much more efficient.
If you have any problem as simple as this, and you are stuck after the using the LCD, I suggest trial and improvement . Clearly when x = 4, we are nearly there . Then do a long division to get the other factor, x-8. Then equate both factors to be equal to zero
It is simple, if we state, that x is ot 0. In this case, we can multipl the equation withh 4x and get thhe quadratic equation. x^2-12x+32=0 which leads to the 2 solutions: x=6+-sqrt(4) x=4 or x=8 We can state, thhat x is not 0, because otherwise the term 8/x is undefined.
Took me about 2 seconds to get 4 as an answer. But to do it on paper, we have x/4 + 8/x = 3. Multiply by 4x to get x^2 + 32 = 12x, or x^2 - 12x + 32 = 0, or (x - 8)(x - 4) = 0, so x = 8 and x = 4 are the answers. This could also be solved using the quadratic formula.
I have to agree with Shelly. When I do one of your problems--I just skip ahead. In the words of my daughter when asked why she didn't want me to help with her math, she told her mother, "Mom, I don't want to know that much."
I was able to get the answer by getting the right hand side of the equation equal to 0, then used the quadratic equation to solve it - got 2 answers, and both worked back in the original problem.
Great little problem, especially making sure that people check the answers actually work. Thanks for creating and sharing. I love your channel. However, I feel that it would be easier for people to understand if, rather than distributing by 4x, the fractions were both converted to 4x denominators then everything multiplied by 4x to produce the equation that becomes the quadratic. As usual with a lot of problems, always more than one way to solve and it’s about finding the method that works for the student.
"How do two numbers added = 3?" Answer, A + B = 3, so 1+2=3, and 2+1=3. So A can be 1 or 2, and B can be 2 or 1. How can x/4 be 1? If x=4. How can 8/x be 2? If x=4. How can x/4 be 2? If x=8. How can 8/x be 1? If x=8. So there they are, 4 and 8.
yes, 4 and 8 are easy and intuitive, but setting up a quadratic in a slightly different way gives x=6+2xsqrt(6) , and 6-2xsqrt(6) for two more solutions.
Greetings. Yes, we can. The answers are X=4, 8. The first thing that has to be done is to get rid of the fractions by multiplying throughout by 4X to get X^2-12X+32=0. Thereafter, we will factorize the equation to get (X-4)(X-8)=0. That is (X-4)=0, (X-8)=0 and X=4, X=8.
I solved this in 10 seconds. Easy. Multiply both sides by 4x, the lowest common denominator. x^2 + 32 = 12x. x^2 - 12x + 32 = 0. (x - 4)(x - 8) = 0. x = 4 or 8
I figured this out in my head. What numbers added together equal 3? 1 and 2. So the fractions have to equal either 1 or 2. So X can be either 4 or 8. Too bad that it takes such roundabout, convoluted methodology in order to prove this.
Answer: X = 8 or x = 4
-------------
I did this problem as follows:
X 8
- + - = 3
4. X
(LCD = 4X)
X^2 + 32 3
------ = -
4X 1
Cross multiplying,
X^2 + 32 = 12X
X^2 -12X + 32 = 0
Factoring,
(X-4)(X-8) = 0
Either X-4 = 0
OR
X-8 = 0
If X-4 = 0,
X = 4
If X-8 = 0,
X = 8
Answer:
X = 4 or X = 8
I think I've solved so many of these this lifetime I could see the answers in front of me. The first obvious one was 4. Because it was a quadratic equation and I instinctively knew the answers were whole numbers figuring out 8 took another 35 seconds (I was timing). Not exactly procedure I'll admit.
Yep and it was super simple
There is a missing exclussion ! Answer : x=4 or x=8 and x not equal 0
@@justme2580
I did this problem as follows:
X 8
- + - = 3
4. X
(LCD = 4X)
X^2 + 32 3
------ = -
4X 1
Cross multiplying,
X^2 + 32 = 12X
X^2 -12X + 32 = 0
Factoring,
(X-4)(X-8) = 0
Either X-4 = 0
OR
X-8 = 0
If X-4 = 0,
X = 4
If X-8 = 0,
X = 8
Answer:
X = 4 or X = 8
To add fractions you need a common denominator like you were taught in 7th grade. To do this make the common denominator 4 * x. Then multiply each term by one to get (x/4 * x/x) + (8/x * 4/4) = 3 to get the equation (x^2 + 32) / 4x = 3 . Then multiply both sides by 4x to get x^2 + 32 = 12x and put equation in standard form of x^2 - 12x +32 = 0 . Then factor to (x - 4)*(x - 8) = 0 to get the answer of 4, 8. There is no need to find the lowest common denominator or to multiply the whole equation my some magic LCD of 4*x*1.
Yes, really not that hard. Dod it in my head.
Really there are several ways of approaching this problem which is all relevant.
@@philipclifton7494, Might not be hard to solve for some of us. Problems like these have more meaning to them than just the solution.
It can be hard for people around 7-12 years old. 12+ should solve it quick with use of LCD and solving ax^2+bx+c=0.... (12x=32+x^2) 😀
@@DarVV This clearly a problem for someone that has had algebra 1 in 9th grade.
You expand 30 second problems MAX into a 20 minute video all the time
Yes it is x=4 and x=8 both work, took me 2 minutes in my head. Step 1 common denominator (4x) x^2/4x +8(4)/4x = 3. Step 2 multiply both sides by 4x, x^2 + 32 = 3(4x) then x^2 + 32 = 12x. Step 3 subtract 12x from both sides x^2 - 12x + 32 = 0. Step 4 factor (x - 4)(x - 8) = 0. Step 5 solve for x in each set x - 4 =0, x - 8 =0. Step 6a add 4 to each side in x - 4 = 0 x - 4 + 4 = 0 + 4, x=4 Step 6d add 8 to both sides in x - 8 = 0, x - 8 + 8 = 0 + 8 x=8 Step 7 test results by substituting results in origional equation. A. 4/4 +8/4 = 3 1 + 2 = 3, 3 = 3,B. 8/4 + 8/8 = 3, 2 + 1 = 3, 3 = 3. Conclusion both 8 and 4 are solutions for x.
The answer is x=4 and/or x=8. It's simple by doing it in steps. The first step is subtract 8/x from each side of the equation, so x/4 = 3 - 8/x The next step is to multiply both sides by 4 so x = 4(3 - 8/x) or 12 - 32/x Then multiply each side by a factor of x. X^2 = 12x - 32 Then isolate the equation to one side. X^2-12x+32=0 This factors out to (x - 4)(x-8)=0 or x=4 and/or x=8
yes I did it that way mentally
@davidseed2939 I remember in school the instructor wanted students to show their work to explain how they arrived at the answer, so the solution was just as important if not moreso than the correct answer(s).
x/4 + 8/x = 3
x +32/x -12 = 0 (times 4)
x² -12x +32 = 0 (times x)
Then just apply quadratic formula
Hi, I'm 75 but still wamt to learn. I was always intimidated by math so I'm trying to conquer my fear. Thank you. Judy from Birmingham
I'm 5 years younger and when I was in school, the minute girls hit teen years, math was for boys, not girls for the most part. If you liked math you were not popular.
I try and try to listen to your video but you go off on a tangent too much. Six minutes into the video and you have not begun solving yet. Would love to learn from you but you just talk and talk and talk and ...................................
He just wasted data space
My thoughts exactly. I fast-forwarded it.
X = 8
Do you not know how to drag the video forward?
@@lamper2 I do, but sometimes he will say something pertinent to me but than he goes off again. Not a problem, just won't watch any of this videos anymore. Anywho I'm retired living comfortably, but I love learning.
I eyeballed it at 4. When you said 2 answers I paused and double checked that 8 would also work.
Same 😊 I was always in advanced math in school and I quickly realized that "I still have it". I also realized that I still have the same problem. I was able to plug in a number by eyeballing it and the number was right but I would not be able to show my work. And I always got deductions for not properly showing my work 😅
I eliminated the denominators and went straight to the quadratic formula, all in my head and it took max 20 seconds...
Yes I think also this is the easiest way
I did the same thing, clearing the equation of fractions, rearranging terms, getting the quadratic.
Actually it was very simple. I remember solving harder problems in 11th grade.
@@vcambridge-ph.d8030 and i bet it didn’t take 20:13 minutes
Me too, this is childs play.
I'm using my UA-cam Premium subscription to learn freestyle swimming, math, Hungarian, Greek, Mandarin Chinese, and other things. But being 72 years old I want each episode to be as succinct as possible without unnecessary talk
He’s TEACHING slow enough for children & teenagers to grasp. I think many of you look at this to test yourself & memory which is not what it’s intended for. Unfortunately, many folks being paid as teachers in public school are not teaching to educate the kids or help them comprehend the information. I feel he has a real grasp on teaching.
Unlike your math teacher, you can pause this. It's not only the speed it is the relentless discursive rambling!
Keep the videos to 5 minutes maximum. I like learning but 16-18 minutes bores me so I have stopped watching/learning.
Only the ignorant are bored!
@@7dimensions-jebname calling is also a common denominator with ignorance…
Followed quite a few of his but i give up screaming here. Too much nervous energy and chat, and oh by the way …..not easy to take notes when he’s going off on digressions and sales pitches in between youtube ads. Too frenetic and nervous chaos.🤯 I’m sure a ton of content you have, but kindly remember kiss. Stop telling us about that and get on with showing us instead of interrupting yourself every 5seconds. I’m going back to the books after this.
Someone who hates math but needs to learn it isn't gonna wanna watch A 20minute video on math@@7dimensions-jeb
I don't even know why... but, I found this challenge so satisfying... I actually remembered... I'm 1966.
Thank you for the confidence boost. Makes me feel like I can help my grandkids.
x/4 + 8/x = 3 -> x^2/4x + 32/4x - 3 = 0 -> ((x/2+32)/4x) - 3 = 0 -> x^2-12x+32 (Quadratic equation)
Given this quadratic equation we can solve for solutions to x using the quadratic formula: (-b+-sqrt(d))/2a where d=b^2-4ac is the discriminant to the quadratic equation:
x = (-b+-sqrt(b^2-4ac))/2a = (-(-12)+-sqrt((-12)^2-4*1*32))/2*1 = (12+-sqrt(16))/2 = 6+-2
The discriminant (16) is greater than 0 so we get 2 solutions:
x1=6+2=8
x2=6-2=4
Intuitively the solution could also have been found by simply looking at the equation x/4 + 8/x = 3 and realizing that probably the solutions are 4 and 8 by substituting into x and doing a bit of simple math: 4/4+8/4=3 as well as 8/4+8/8=3. With more complex equations and/or unintuitive values of x the best method is to use formula for the discriminant and the quadratic formula.
I followed and loved you solved the algebra!!!!
You drag it out too long.
He babble too much
@@eugenewilliams8440,
he asks you and me to understand that talking is much more amusing than algebra .
Two of the people who replied to the above comment need refresher courses in English.
It would give Young Sheldon pause.
1)- Thank you for taking the time, but you took us around the world and brought us back to the beginning.
2) - Algebra is not as easy as you have tried to teach your pupils or students.
3) recommend you to teach your skills in an easy meeting!!!!.
You should always use the quadratic equation. When I discovered the quadratic equation I was mad that math teachers wasted my time with factoring.
Agreed! Factoring is trial and error guess-work - just hoping that known factors of 32 just happen to add up to a nice round number like 12. 40 years of working with computers tells me that a simple function works best: x = solve_quadratic(1,-12,32) - sorted! 🤓
X=4 ie 4x( x/4 +8/x) = 4x × 3
ie x.x + 32 = 12x
x.x + 32 - 12x = 0
X.X - 8X - 4X + 32= 0
X(X-8) - 4(X-8)=0
(X-8)(X-4=)=0
Hence X = 8 or 4
To solve the equation x/4 + 8/x = 3, we can first multiply both sides of the equation by 4x to eliminate the fractions.
(x/4 + 8/x) * 4x = 3 * 4x
x^2 + 32 = 12x
Next, rearrange the equation to set it equal to zero:
x^2 - 12x + 32 = 0
Now, we can factor this quadratic equation:
(x - 8)(x - 4) = 0
Setting each factor equal to zero gives us the possible solutions for x:
x - 8 = 0 or x - 4 = 0
x = 8 or x = 4
Therefore, the solutions to the equation x/4 + 8/x = 3 are x = 8 or x = 4.
This is the way I solved it, except just using the quadratic formula once it's in standard form. x = (-b ± sqrt(2ac))/2a, where a=1, b=-12, c=32
taught math to most of my engineering class for 4 years. Went to work and became a digital designer who uses almost no math. I am 77 and love relearning, thank you!
4 was the first number that made any sense in my mind and without having much of an understanding of algebra or the other methods used to solve this problem.
I went for 8
Relatively simple proof of quadratic formula:
ax² + bx + c = 0
(ax² + bx + c) /a = 0 /a (a≠0) (simplify x²)
x² + bx/a + c/a = 0
x² + bx/a = -c/a (isolate x terms on left)
x² + x•b/a + b²/4a² = -c/a + b²/4a² (completing the square on left by taking 1/2 of b/a = b/2a and squaring it, and adding to both sides)
x² + bx/a + b²/4a² = -4ac/4a² + b²/4a² (combining terms on right)
x + b/2a = ±√(b² -4ac) / √4a² (taking square root of both sides)
x + b/2a = ±√(b² -4ac) / 2a
x = -b ±√(b²-4ac) / 2a
Guess and check is quickest in this case. I need 2 numbers that add up to 3. Hmmm 1+2=3. Looks like 4 works in that case. Algebra is not always needed, sometimes a little number sense can work.
I did that too, but stopped at 4.
A little more persistence could have given "What about 8?", since 8/8 is 1, and 8/4 is 2, and 2+1=3. Once you see it explained, it seems too simple.
Second degree quadratic equation : ax^2 +bx +c = 0
Delta = b^2 - 4ac
X1 = (-b + √(Delta) ) / 2a
X2 = (-b - √(Delta) ) / 2a
So with x^2 - 12x +32 = 0
a=1, b=-12, c = 32
Delta = 16
X1 = 8
X2 = 4
I am happy to inform you that your lessons have produced solid results.
6 months ago I would have given up on this one and now I can solve it using algebra.
Now I'll watch the video to see if my method was the right one but I'm confident it was. :)
Thanks!
Can't you just look at it, and it is 100% obvious that if you sub in x=4, it will work?
@@marscience7819 Yeap! But if you want to deal with tougher equations you need to learn algebra. In fact, I did not even think about plugging in the number 4 when I saw this problem.
In other words I went from thinking like you did- totally fine- to thinking algebra first.
So, I stand corrected in the sense that I would not have given up on the problem but I don't think the same way since I worked on my algebra skills which are still a work in progess BTW. ;)
Try this one and see how you are doing without algebra: x/89 + 143/x = 542
Just a hint: number 4 won't work. ;)
I solved this intuitively, and now that I’ve watched the video I’m sure I couldn’t solve it. I give up.
Thank you for your service. I was denied the honor in 1976 due to being an Epileptic, but so grateful to those that can and do
(x/4)+(8/x)=3
therefore x not= 0
x × {(x/4)+(8/x)=3}
x^2/4+8=3x
4×{x^2/4+8=3x}
x^2+32=12x
x^2-12x+32=0
note => 2 solutions for x but not 0
the quadratic equation:
[-b +/-sqrt(b^2-4ac)]/2a
for
x^2-3x+8=0
a=1
b=-12
c=32
so
[-b +/-sqrt(b^2-4ac)]/2a
[12+/-sqrt(144-128)]/2
[12+/-sqrt(16)]/2
[12+/-4]/2
sol.1 x=4
sol.2 x=8
verify
(x/4)+(8/x)=3
sol.1 x=4
4/4 +8/4=?3
1+2=❤3✔️
sol.2 x=8
8/4+8/8=?3
2+1=❤3✔️
Gosh u r quick!
Beat me to the answer set:-D
X =4 or 8. Multiply through by 4x and you'll have a quadratic so make it equal zero and use the quadratic formula to solve
@@tomtke7351 Fast and accurate
You are bringing back lots of teaching I have had. My first algebra class was 1965!!! Thank you!!
This was a good one. I labored for a long time before finally realizing that I had to get that x out of the denominator. So here comes the quadratic equation, and it was simple to solve.
Could you just have looked at it, and it is obvious that if you sub in x=4, both sides are gonna be the same?
@@marscience7819 Yeah, I kind of did a facepalm when I got the answers. Sure, one number plus another number equals 3. The first choice would be that one fraction is 1 and the other is 2. That would have been so easy. But I knew this was an algebra problem, so I tried to use algebra to solve it.
@@marscience7819and you missed the other value. there's actually two values for x.
I really like your video. You are very thorough and try to point out the complexities by referring to many points of solving the equations. Yes it may seem tedious but it helps me. I still have some trouble knowing when I need to look for lowest common multiplier versus lowest common denominator. I think it takes a lot of patience and determination to get good at this stuff. I am a biologist and though I've had some training in math, I was always kind of slow, but usually I could get through the courses with a good grade. Now I am retired and I find it a fun game to redirect my attention to algebra and other math, so I hope you and I will keep enjoying the challenges. Keep up the good work, and best wishes!
No need for complicated algebra. Use cross reduction and you get 1+2=3. The X’s cancel each other out, 4 and 8 a divisible by 4 leaving you with 1 over 1 and 2 over 1.
"Use cross reduction"
That only works *multiplying* fractions. This is addition. Anyway, what is wanted IS the X and you eliminated it completely.
@@thomasmaughan4798 so, I simply solved the equation. So, you’re right I didn’t solve for X. However, cross reduction is a form of simplifying fractions to their lowest form and does work for addition and subtraction of fractions.
@@potterspride1 Well, whatever. I trust you do not build bridges.
@@thomasmaughan4798 no, I don’t build bridges. However, if I did, I wouldn’t be using algebra to build it. I would use exact numbers, and if a number wasn’t known, I would go out and measure it again myself.
@@potterspride1 "no, I don’t build bridges."
That's a relief!
I took algebra in my first year in high school as a freshman. First we find the common denominator, which is 4x. Then, we multiply the 4x both sides which gives us x^2 + 32 = 12x. Then we make the equation to equal 0 by subtracting 12x both sides, which is gives us a quadratic equation of x^2 - 12x + 32 = 0. Thirdly, we factor the trinomial into two binomials using the FOIL method which is (x-4)(x-8) = 0. Finally we make each equation to equal 0, which is x-4=0 and/or x-8 = 0 and then solve for x. We get x = 4, x = 8.
The key to solving is to get all of the terms under a common denominator. The result is a quadratic equation in the numerator, which is easily factored into the two solutions.
Exactly.
I had 2 years of math back in the 50s !. I’m so glad the way I learned algebra was not as complicated. I resolved this problem in seconds.
I had to retake an algebra course in my late thirties since it was a pre-requisite to a university certificate I was taking. I am now 67 and I solved the problem like you in seconds. He had me doubting myself since he taking so long to explain it.
Interruptions of explanations to promote other programs is unhelpful.
Well, I did it by taking care of the fractions first and then used the cross product to solve the problem.
At the end I came up with the right equation anyway i.e. x Squared + 32 = 12x
Then I substracted 12x from both sides, factored and came up with (x -4) (x - 8) = 0 and voilà!
But I could have done it by multiplying the whole equation by 4x as you did.
When I see fractions and an (=) sign I always think cross product. ;)
This particular equation can also be factored into 2 Binomials sq. by each other
(x-4)*(x-8)=0 and {x = 4; x = 8}
x^2 - 12x + 32 = 0
x/4 * 4x/1 + 8/x * 4x/1 = 3 * 4x
fours cancel out, and leaves you with x^2; 8/x * 4x/1... the x's cross-cancel and leaves you with 8 * 4 = 32;
Then, whatever you do to one side always do it on the other side of the "=".
that gives you 12x
quadratic equation: Ax^2 + Bx + C = 0 (in standard form)
or (-)b±√b^2 - 4*a*c/2*a
X=4 & 8.
Did it in my head & within a few minutes.
This one is easy by inspection. 1+2=3, 2+1=3
Ya'll stop complaining about him being precise in his teaching. The classical style demands he not leave out a step
Super simple, I would expect a c student to get this right in less than 20 seconds.
1:) Bow tie method:
(x² + 32) / 4x = 3
2:) get rid of fraction by multiplication:
[(x² + 32) / 4x ] * 4x = 3 * 4x
x² + 32 = 12x
3:) set equation equal to zero and solve via foil
x² -12x + 32 = 0
(x-8)(x-4) = 0
x = 8 and x = 4.
Like I said super simple if you know what you're doing.
x/4 + 8/x = 3
Let t = x/4
t + 2/t = 3
t^2 -3 t + 2 = 0
(t -1)(t-2) = 0
x = 4 t
x = 4
Or
x = 8
This is the way I did it and it seems to simplify things. Noticing the 2nd term is 2/t if the first is t is a nice simplification.
This is always where I got in trouble in school. They always wanted to show the work. They wanted to see how we solved it. That takes too long. I just look at the problem and my brain figures out the answer. This happens almost instantly
Thank you for your channel. I always loved math and now use your channel for brain exercises (I am 74). Shoutout to all my math teachers. I went as far as calculus -
Mental math 101. Look, think, solved it! Something + something = 3. Answer: 4 😊
Or 8, it's a quadratic
Correct. Because 4/4 + 8/4 = 1 + 2 = 3
If X squared over 4X + 32 over 4X = 3, then cross multiplication would mean that X squared + 32 = 3(4X), simplified to X squared + 32 = 12X. Then X squared minus X + 32 = 0. It then becomes (X-4)(X-8)=0. X-4=0, then X = 4. X - 8 = 0, then X = 8.
I solved it in one sec, just looking at the thumbnail as the notification of the video appeared.
The general case is:
(x / a) + (na / x) = (n + 1)
That can be solved by applying this formula:
x = (a / 2) [(n + 1) ± (n - 1)]
Which is:
S: { a ; na }
got it 4 & 8 solved by sight easy calculation had to be 2 + 1 and 1 + 2 thanks for the fun
yea it was real easy
It's simple enough, that I don't need to create the full quadratic equation. The mean of the roots is 3 (constant term) * 4 (recibrocal of linear term) / 2 = 12/2 = 6. Obviously x = 4 and x = 8 result in 1+2 or 2+1, both are 3. Or you calculate 8 (1/x term) * 4 (see above) = 32 = x_1 * x_2. Or you take the long way with x^2 - 12 x + 32 = 0 => x = -(-12)/2 +/- sqrt(36 - 32) = 6 +/- 2.
Isn’t it 4 without even any solving? 😂😂
By multiplying 4x everywhere you get
X2+32=12x
(X-4)(x-8)=0
X=4
X=8
You explain to much which deviate from yr target.we don't have too much time to explain the problem.Pse shorten the explanation.
Please remember there are others here who are not as knowledgeable as others. Have patience with the slower ones, ok. This is a how teaching goes.
What you are seeing is the schtick. He is an elite mathematician and wants to give you a real math experience. The "prollem" looks simple and turns out it is rife with pitfalls. You, therefore, can't solve it and he steps in step by step and solves it. This is how you learn math. If you want to play childhood T-ball, this is not the guy. He's gonna test you and, frankly, I like that. He also designs tests for a living, so he's going to know how folks get sidetracked. That is valuable information! Love, love, love this dude!
(x/4)+(8/x)=3
=>(x^2+32)/4x=3
=>x^2+32=12x
=>x^2-12x+32=0
=>x^2-4x-8x+32=0
=>x(x-4)-8(x-4)=0
=>(x-4)(x-8)=0
=> x=4,8
I didn't bother to watch more than the first few minutes. It takes about a minute to solve it
(x/4)+(8/x)=3
Domain: x≠0
4x(x/4)+4x(8/x)=4x(3)
x²+32=12x
x²-12x+32=0
(x-8)(x-4)=0
x-8=0
x=8 ❤
x-4=0
x=4 ❤
Without watching the video, this one is quite easy. You are allowed to multiply both members by any value without changing the equation. Let's do it by x. Which gives us 1/4 x^2 -3x +8 =0.
∆=b^2-4ac = 9 - 8 = 1.
x1&2 = (-b +- √∆)/2a = (3 +- √1)/1/2 = 8 & 4.
Thank you for taking the time to explain! I am enjoying the learning process!
formula for roots of a quadratic equation X**2 -12X + 32 = 0 (which we get when multiplying each element with 4X) gives us 8 and 4 as 2 roots of X
Multiply by 4x to eliminate denominators.
You get a quadratic; apply the quadratic formula.
Or, even better, in this case the quadratic easily factors into a product of linear factors
(x-4)(x-8).
Been many years for me since school. Thank you.
I think I knew this sinds 1962: solution x =4 and x=8; by writing first part as the sum with the common denominator 4x and multiply by 4x you get the quadratic equation:ax^2+bx+c= x^2-12x+32=0; D=b^2-4ac; x=(-b±√D)/2a (D is the Discriminant)
you did it the hard way with the -b part, but yes technically that is correct. Now, if the test giver had requested foil or factoring, then no this would be wrong.
After 50+ years of taking algebra, I got this with no problem. Easy peasy.
I had 4 and 8 on sight. I am a 61 year old British electronics engineer who mainly uses standard equations, calculus and FFT.
The answer is simply x=4 and x=8
Simply find the common denominator if you have the denominator of 4x multiply to both numerator and do the process of solving equations..
So x squared + 32 =12x
So get the value of x it should be x=4 and x=8
辺々4xを掛けx^2+32=12x
12xを移項し(x-4)(x-8)=0
∴x=4,8
Simplify the equation to the standadrd quadratic x^2 -12x+32=0; factorising you get (x-8)(x-4)=0. Hence X=8 or 4.
"Simplify the equation"
Increasing the order (from x to x^2) is not simplification. your mileage appears to vary.
x=8
8/4=2 + 8/8=1
2+1=3
(Edited to add):
x could also be 4
4/4=1 + 8/4=2
1+2=3
This problem is EXTREMELY SIMPLE!! The result MUST equal 3, and "X" must be the same number. In this case "X" = 4. So, 4/4 = 1 + 8/4 = 2 Guess what? 1 + 2 = 3 Kindergarten wasn't wasted on me.
😂😂😂
I multiplied by 4x at the first step for convenience, brought all the terms to one side and factored. Took 3 or 4 lines to the solution.
X/4 + 8/X = 3
Muktiply both sides by 4X
Gives
X² + 32 = 12X
X² - 12X + 32 = 0
(X - 8) (X - 4) = 0
X = 8 or X = 4
Inspection shows x=4 is one solution, with the 2nd term larger than the 1st. Increasing x makes the 1st term larger than the 2nd, so x=8 is another solution. Check video. Eventually found both answers after 17 min. Jeez! 3,500 people actually liked this!
x/4 -3 = -8/x ==> 3x/12 -36/12 = -8/x (3x-36)/12= -8/x ==> 3x**2-36x=-96 3x**2-36x+96=0 (x-8)(x-4)=0 x=4, 8 it is very easy to solve it
X/4 + 8/x = 3
X^2/4x + 32/4x = 3
(X^2 + 32)/4x = 3
X^2 + 32 = 12x
X^2 - 12x + 32 = 0
(X-4)(x-8) = 0
X = 4 , 8 ✅
Because of algebra I know that x will be the same number and that there might be more than one choice. Basic knowledge of fractions told me that if I could get fractions equal to 1 and 2 it would work because 1+2=3 and 2+1=3. 4/4=1 and 8/4=2 so one solution is 4. 4/8=2 and 8/8=1 Therefore the second solution is 8.
I multipled both sides of the equation by x to get the variable out of the denominator. Then you get a quadratic equation. I moved everything to one side of the equation so it’s equal to 0. Then I factored to get (x-8)(x-4)=0. So the solutions are 8 and 4.
When solving equations you move terms from one side of the equals sign to the other and change the sign. Positive terms become negative, division becomes multiplication etc it is much more efficient.
So solve for X and show your work.
too much yapping, skip til 5:30
Even after 5 30 there's too much yapping.
If you have any problem as simple as this, and you are stuck after the using the LCD, I suggest trial and improvement . Clearly when x = 4, we are nearly there . Then do a long division to get the other factor, x-8. Then equate both factors to be equal to zero
It is simple, if we state, that x is ot 0. In this case, we can multipl the equation withh 4x and get thhe quadratic equation.
x^2-12x+32=0
which leads to the 2 solutions:
x=6+-sqrt(4)
x=4 or x=8
We can state, thhat x is not 0, because otherwise the term 8/x is undefined.
50 years ago, while in high school and college. Had no problem solving math problems like this one. There is no reason now...
Took me about 2 seconds to get 4 as an answer. But to do it on paper, we have x/4 + 8/x = 3. Multiply by 4x to get x^2 + 32 = 12x, or x^2 - 12x + 32 = 0, or (x - 8)(x - 4) = 0, so x = 8 and x = 4 are the answers. This could also be solved using the quadratic formula.
I have to agree with Shelly. When I do one of your problems--I just skip ahead. In the words of my daughter when asked why she didn't want me to help with her math, she told her mother, "Mom, I don't want to know that much."
I came up with 4 in a matter of seconds, when you said two solutions I came up with 8 quickly
I was able to get the answer by getting the right hand side of the equation equal to 0, then used the quadratic equation to solve it - got 2 answers, and both worked back in the original problem.
Gleichung mit 4x Multiplizieren --> x^2 - 12x + 32 = 0 Quadratische Ergänzung ergibt --> ( x - 6 )^2 = 4
Replace the x with 4, solve 4/4=1; solve 8/4= 2; add 1+2=3 simple
Great little problem, especially making sure that people check the answers actually work. Thanks for creating and sharing. I love your channel.
However, I feel that it would be easier for people to understand if, rather than distributing by 4x, the fractions were both converted to 4x denominators then everything multiplied by 4x to produce the equation that becomes the quadratic. As usual with a lot of problems, always more than one way to solve and it’s about finding the method that works for the student.
x/4 + 8/x = 3
x²/4x + 32/4x = 3
x² + 32 = 12x
x² − 12x + 32 = 0
(x − 4) (x − 8) = 0
x₁ = 4 ∨ x₂ = 8
𝕃 = {4, 8}
Great content. How about just 1 tangent per video?
"How do two numbers added = 3?"
Answer, A + B = 3, so 1+2=3, and 2+1=3.
So A can be 1 or 2, and B can be 2 or 1.
How can x/4 be 1? If x=4.
How can 8/x be 2? If x=4.
How can x/4 be 2? If x=8.
How can 8/x be 1? If x=8.
So there they are, 4 and 8.
yes, 4 and 8 are easy and intuitive, but setting up a quadratic in a slightly different way gives x=6+2xsqrt(6) , and 6-2xsqrt(6) for two more solutions.
Y=X/4， 2/Y=8/X，Y+2/Y=3， Y^2-3Y+2=0，（Y-1）（Y-2）=0， Y=（1，2）， X=（4，8）
Greetings. Yes, we can. The answers are X=4, 8. The first thing that has to be done is to get rid of the fractions by multiplying throughout by 4X to get X^2-12X+32=0. Thereafter, we will factorize the equation to get
(X-4)(X-8)=0. That is (X-4)=0, (X-8)=0 and X=4, X=8.
I solved this in 10 seconds. Easy. Multiply both sides by 4x, the lowest common denominator. x^2 + 32 = 12x. x^2 - 12x + 32 = 0. (x - 4)(x - 8) = 0. x = 4 or 8
Time for me was only about 4 or 5 minutes, but I'm slow. Still this was not complicated. Do not need anywhere near 20 minutes to explain this one.
Thank you, I am about to take college algebra and am a total beginner
I first tried to insert 8 for x and it worked! XD
You could even replace the '8' with an x in the problem.
I instantly divided into c (each x turning into 1.
1x8=8
1x4=4
=8/4
=2
1+2=3
I figured this out in my head. What numbers added together equal 3? 1 and 2. So the fractions have to equal either 1 or 2. So X can be either 4 or 8. Too bad that it takes such
roundabout, convoluted methodology in order to prove this.
The easiest and BEST way with answer is @18:00 mark. Simple and takes 10 seconds to get the answer instead of 20 minutes.
4 and 8