Unless I missed it, there's a rather important bit missing here: for two particular values of x, the expression in undefined because the denominator is 0. Anyone whose algebra skills extend to being able to factor x-x² to x(1-x) should be able to quickly see that the denominator equals -1 × the numerator so the answer must be -1. For finding out the answer, factoring is way more complexity and faff than you need. But the one reason it might be worth doing is to help see what values of x give 0 in the denominator. Again, anyone who can factor x²-x to x(x-1) should be able to quickly see that for x=0 and x=1 the denominator is zero. So the answer to the question is -1, provided than x≠0 and x≠1 You went through all the bother of factoring and then didn't actually use the factored version of the expression for the one thing it's useful for!
Just substituting the first two answers leads to division by zero which is undefined. So it isn't those answers. As you say the answer is -1 for all x except x =E {0,1}
That was unnecessary. Just thinkk about, what --1*(x-x^2) is... (-1)*(x-x^2)=((-1)*x-(-1)*x^2)=(-x--(-x^2))=(--x+x^2)=(x^2--x) which is the same as the Denomminator. So the result is -1 (for every x for which x^2-x is not zero, i.e. for every x except x=0 or x=-1).. For x=0 or x=-1, the term is undefined, because thhe denomminator is 0 in that cases..
Factoring out x in top and bottom gives: x(1 - x)/x(x - 1) x in num cancels x in denom leaving: (1 - x)/(x - 1) = -(x - 1)/(x - 1) x - 1 in num cancels x - 1 in denom leaving -1. Answer is: -1 =
MANY MISS THE POINT! The fundamental, primary issue before a student begins to work on this problem is to understand what the goal is. If this problem is given as homework where "show your work" is a necessity, then the explanation and calculation given by John Mr. Mathman is right on. HOWEVER, if this were a question from an SAT or similar standardized "fill the oval with a dark mark from a No. 2 lead pencil" test, the analysis shown would be a counterproductive and an unnecessary waste of time. On standardized tests the ONLY goal is getting the right answer, nothing else. For certain questions - like this one - the "random substitution" method works best: Pick a random number, plug it in for the variable, do the calculation, and compare your answer to the answer choices provided. If your calculated answer corresponds with ANY answer choice, that choice MUST be the right one. Done and done. If you distrust the method and have time, you can run another random number which should confirm your original answer. Since you're not charged with demonstrating a "proof" but, rather, only finding the answer, the easiest and quickest method is superior. If the goal, however, is to provide an opportunity to show your teacher that you understand HOW to derive correct answers, then the long and formal way is necessary. So I chose 13. Pretty random, and amenable to easy arithmetic calculations. So.... (13-13^2) / (13^2-13) = (13-169)/(169-13) = (-156)/(156) = -1. Since -1 IS the answer for 13, it must also be the answer for all numbers; no other answer choice satisfies 13, so all of the remaining choices are wrong. A nervous Nellie might confirm with a different number. Try -1/2. You get: [(-1/2) - (-1/2)^2] / [(-1/2^2) - (-1/2)] = [(-1/2) - (1/4)] / [(1/4)-(-1/2)], or (-3)4) / (3/4) which is -1. Written using decimals is easier to see: [(-0.5) -(-0.5^2)] / [(-0.5^2) - (-0.5)] = - 0.25 / 0.25 = -1. Same answer. The right one, whose acquisition is the only goal. Easy peasy and a real timesaver.
I completely agree that if the requirement is only to provide an answer then just get to the answer the quickest way you can. But I don't think substitution is quickest here. Quickest is to immediately observe that the denominator is the negative of the numerator so the answer must be -1.
original form: y = (x - x²) / (x²-x) 1) we have a GCF of x so: y = [ x (1 - x) ] / [ x ( x - 1)] 2) we have 1 - x in numerator and x - 1 in denominator so lets write it so they are exactly the same inside the ( ): y = [ x (1 - x) ] / [ - x (1 - x)] 3) Now we can cross cancel as the ( ) portion is exactly the same this gives us: y = x / - x this means y must equal -1 proof of step 2: y = x ( 1 - x ). This equals the original numerator of (x - x²) y = -x ( 1 - x). This equals -1x + x². If you remember that a - b is the same as -b + a then we know that -x + x² is the same as x² - x. This means the denominator is the original denominator. Super simple if you remember and fully understand the rules of addition and subtraction when it comes to numbers with opposite signs.
For X≠1 and X≠0 : -1. The proposed answers are not complete as they don't include the undefined cases. IF X=0 OR X=1, then, you get 0/0. That's undefined or Not a Number (NaN)
x - x^2 / x^2 - x So let’s start out by realizing that if x is zero then this is undefined so x cannot be zero We can alter the denominator and the numerator I am going to multiply the numerator by -1 so -1 * (x - x^2) = -1(-x + x^2) = -1( x^2 - x) -1* (x^2 -x) / (x^2 - x) = -1 So looking up you have -1 as the answer But is this the only way to solve the equation We can move 1x from the equation. (1-x)/(x-1) = k (1-x) = k(x-1) 1- x - kx + k= 0 1 + k - x(1+k) = (1-x)(1+k)=0 solutions are x=1, k = -1
too much work just realize the numerator can match the denominator by multiplying it by -1. Then you can cross out the terms in ( ). Bobs your uncle -1 divided by 1 and even a D student knows that is -1. I am a fast typist and I could do the work 10 times in my head before typing the first sentence.
Before watching. Factor x out top and bottom. Look at the denominator and determine the domain. X cannot be 0, or 1, or one would end up dividing by 0, which without a lim in front is a no-no. That leaves answers c and d, so 50% chance to guess correct from here. Now, since we can now cancel the x/x part, playtime starts. Is there anything to be done to divide (1-x)/(x-1)? Yes, multiply top, or bottom, not both 😃, by -1, and switch signs in the brackets to compensate for the introduction of the -1. Now the answer should be clear since one ends up with -1/1 or 1/-1, both result in -1, I.e the answer should be c. Now let’s watch🤓after watching, yes, if you have no clue, guess, but know how the teacher/prof scores. I had one prof who would deduct points for wrong answers. This changes the math on guessing a lot! And I love that I guessed the term ‘no-no’ 😎 but, I disagree that one cannot/ should not cancel the x/x at the earliest opportunity. No need to drag terms around beyond when they can just ‘go away’.
John, It seems that, in the second part of the video, everything gets cancelled. But I can't see the result being 1. You crossed the whole fraction. Cansomeone help me with this,please? Thanks
Not sure if you still need help, but if so I hope this helps. Do you understand that 4 - 5 is the same thing as -5 + 4. If so, then apply that concept to the numerator. Now the numerator is -x² + x. We want that to be exactly the same as the denominator which we can do by multiplying by -1. (We are NOT changing the values of x, just the signs of subtraction or addition.) So we now have -1(x² - x). The denominator is really the same thing as 1(x²-x). so Cross out the terms inside the () and you are left with -1 divided by 1.
The answer is “None of the above”. This is still a function of X so it depends on what you plug in for X. The answer is -1 for most values of X and “Undefined” for x = 0 or x = 1.
They're nothing to "plug-in" as x at the end, the xs cancel. I got -1 in my head in about 10 seconds. This guy drags it out. You just multiply by -1/-1 and only process the numerator first and you end up cancelling the original numerator and denominator and are just left with the factor 1/-1 that you held back and this = -1.
Simplifying this equation involves factoring out X in both the numerator and the denominator an then dividing. In most cases X/X = 1 and can be dropped. This is not true when X happens to be 0. This example is interesting ibecause X completely disappears from the simplified function. The problem does not provide information on how to handle discontinuities so the domain of the original function must apply to the domain of the simplified result.
That only proves the special case for x=2. We need the answer for any possible value of x. Although given that this is a multiple choice question, if we assume that one of the options is going to be correct then your approach is sufficient.
I think you over complicated the algebraic solution. Setting aside the fact that x = {0,1} yields an undefined solution, since that was not one of the multiple choices, for all other real number values of x, (1-x)/(x-1)=-1.
-1 in about three seconds. Seriously? "MANY WILL GET WRONG" crap again? Can't you just teach without baiting? It's demeaning to those who are struggling. This is the reason people say, "Math is hard!" You put them down before they ever get the chance to learn. Telling them they're just part of the group who get the problem wrong only reinforces their anguish. Be human. Stop with that "MANY WILL GET WRONG" crap. Teach the math.
You are a teacher with 630k students, so you have a responsibility to pronounce math terminology correctly. It's algebraic, not algebaric, so it's pronounced al-juh-bray-ic, not al-juh-bare-ic.
c) -1 AND a) 0. I have to disagree with the basic rule that a fraction's denominator can never be zero. I hereby declare that it CAN be zero when the numerator is also zero. So 0/0 =0. In this special case, you CAN divide by zero because nothing divided by nothing is nothing. That is intuitively correct. If the numerator is any number other than zero, the rule is true. Therefore answer a) 0 is also correct, when X = 0.
@@terry_willis Division by 0 is ∞ and it can be proven. All you have to do is evaluate the following: lim (1/x) x>>0 Now start X at say 1 This evaluates to 1 Now set x = .00001 This evaluates to 100,000 So as x approaches 0, 1/x approaches ∞ So division by 0 has no finite answer.
One for the kiddies: If you take 3 smarties and divide by nothing what do you have? STILL 3 SMARTIES! (.munch.munch. ☺️😋) If it's -2 outside and the weather man says it is going to get warmer by one-half, what is the temperature going to be? -1 🎉. TA-DA! 🎉
Unless I missed it, there's a rather important bit missing here: for two particular values of x, the expression in undefined because the denominator is 0.
Anyone whose algebra skills extend to being able to factor x-x² to x(1-x) should be able to quickly see that the denominator equals -1 × the numerator so the answer must be -1.
For finding out the answer, factoring is way more complexity and faff than you need. But the one reason it might be worth doing is to help see what values of x give 0 in the denominator. Again, anyone who can factor x²-x to x(x-1) should be able to quickly see that for x=0 and x=1 the denominator is zero. So the answer to the question is
-1, provided than x≠0 and x≠1
You went through all the bother of factoring and then didn't actually use the factored version of the expression for the one thing it's useful for!
Just substituting the first two answers leads to division by zero which is undefined. So it isn't those answers. As you say the answer is -1 for all x except x =E {0,1}
c) = (-1) is correct because the numerator is the negative of the denominator.
Got it right! I just plugged in a couple numbers and it came out to -1 both times.
@pysankar Same here! Basic maths and a little bit of algebra.
That was unnecessary. Just thinkk about, what --1*(x-x^2) is...
(-1)*(x-x^2)=((-1)*x-(-1)*x^2)=(-x--(-x^2))=(--x+x^2)=(x^2--x) which is the same as the Denomminator. So the result is -1 (for every x for which x^2-x is not zero, i.e. for every x except x=0 or x=-1).. For x=0 or x=-1, the term is undefined, because thhe denomminator is 0 in that cases..
Factor top to -1(x^2 -x), cancel with bottom leaving -1
A 2 second mental solution
X-X^2 = -1( X^2 - X )
The X^2 - X cancels out leaving -1
However, X=1 and X=0 are excluded because the denominator is 0.
Well, there is a rule: A - B == - (B - A). This can be used to great effect.
I always knew it as A - B == -B + A. Which is of course the same thing but doesn't use ().
Factoring out x in top and bottom gives:
x(1 - x)/x(x - 1) x in num cancels x in denom leaving:
(1 - x)/(x - 1) = -(x - 1)/(x - 1)
x - 1 in num cancels x - 1 in denom leaving -1.
Answer is: -1
=
MANY MISS THE POINT! The fundamental, primary issue before a student begins to work on this problem is to understand what the goal is. If this problem is given as homework where "show your work" is a necessity, then the explanation and calculation given by John Mr. Mathman is right on. HOWEVER, if this were a question from an SAT or similar standardized "fill the oval with a dark mark from a No. 2 lead pencil" test, the analysis shown would be a counterproductive and an unnecessary waste of time.
On standardized tests the ONLY goal is getting the right answer, nothing else. For certain questions - like this one - the "random substitution" method works best: Pick a random number, plug it in for the variable, do the calculation, and compare your answer to the answer choices provided. If your calculated answer corresponds with ANY answer choice, that choice MUST be the right one. Done and done.
If you distrust the method and have time, you can run another random number which should confirm your original answer. Since you're not charged with demonstrating a "proof" but, rather, only finding the answer, the easiest and quickest method is superior. If the goal, however, is to provide an opportunity to show your teacher that you understand HOW to derive correct answers, then the long and formal way is necessary.
So I chose 13. Pretty random, and amenable to easy arithmetic calculations. So....
(13-13^2) / (13^2-13) = (13-169)/(169-13) = (-156)/(156) = -1. Since -1 IS the answer for 13, it must also be the answer for all numbers; no other answer choice satisfies 13, so all of the remaining choices are wrong.
A nervous Nellie might confirm with a different number. Try -1/2. You get: [(-1/2) - (-1/2)^2] / [(-1/2^2) - (-1/2)] = [(-1/2) - (1/4)] / [(1/4)-(-1/2)], or (-3)4) / (3/4) which is -1.
Written using decimals is easier to see:
[(-0.5) -(-0.5^2)] / [(-0.5^2) - (-0.5)] = - 0.25 / 0.25 = -1.
Same answer. The right one, whose acquisition is the only goal.
Easy peasy and a real timesaver.
I completely agree that if the requirement is only to provide an answer then just get to the answer the quickest way you can.
But I don't think substitution is quickest here. Quickest is to immediately observe that the denominator is the negative of the numerator so the answer must be -1.
and this explains why multiple choice has no place in mathematics unless work is shown to prove that you didn't just "luck your way into it".
original form:
y = (x - x²) / (x²-x)
1) we have a GCF of x so:
y = [ x (1 - x) ] / [ x ( x - 1)]
2) we have 1 - x in numerator and x - 1 in denominator so lets write it so they are exactly the same inside the ( ):
y = [ x (1 - x) ] / [ - x (1 - x)]
3) Now we can cross cancel as the ( ) portion is exactly the same this gives us:
y = x / - x this means y must equal -1
proof of step 2:
y = x ( 1 - x ). This equals the original numerator of (x - x²)
y = -x ( 1 - x). This equals -1x + x². If you remember that a - b is the same as -b + a then we know that -x + x² is the same as x² - x. This means the denominator is the original denominator.
Super simple if you remember and fully understand the rules of addition and subtraction when it comes to numbers with opposite signs.
Glad to see someone was taught the right way like I was.
Thanks Mr Mathman. The channel helps me remember how to learn. Thankyou for the effort.
Thank you john
For X≠1 and X≠0 : -1. The proposed answers are not complete as they don't include the undefined cases.
IF X=0 OR X=1, then, you get 0/0. That's undefined or Not a Number (NaN)
Practice Practice Practice, very informative video
x - x^2 / x^2 - x
So let’s start out by realizing that if x is zero then this is undefined so x cannot be zero
We can alter the denominator and the numerator
I am going to multiply the numerator by -1 so
-1 * (x - x^2) = -1(-x + x^2) = -1( x^2 - x)
-1* (x^2 -x) / (x^2 - x) = -1
So looking up you have -1 as the answer
But is this the only way to solve the equation
We can move 1x from the equation.
(1-x)/(x-1) = k
(1-x) = k(x-1)
1- x - kx + k= 0
1 + k - x(1+k) = (1-x)(1+k)=0 solutions are x=1, k = -1
too much work just realize the numerator can match the denominator by multiplying it by -1. Then you can cross out the terms in ( ). Bobs your uncle -1 divided by 1 and even a D student knows that is -1. I am a fast typist and I could do the work 10 times in my head before typing the first sentence.
You're a great teacher.
X^2 - X = -1 * ( X - X^2) in the denominator. It cancels out leaving 1 / -1
Yay! I got it right! Love your challenges!❤🧮➖➕
Before watching. Factor x out top and bottom. Look at the denominator and determine the domain. X cannot be 0, or 1, or one would end up dividing by 0, which without a lim in front is a no-no. That leaves answers c and d, so 50% chance to guess correct from here. Now, since we can now cancel the x/x part, playtime starts. Is there anything to be done to divide (1-x)/(x-1)? Yes, multiply top, or bottom, not both 😃, by -1, and switch signs in the brackets to compensate for the introduction of the -1. Now the answer should be clear since one ends up with -1/1 or 1/-1, both result in -1, I.e the answer should be c. Now let’s watch🤓after watching, yes, if you have no clue, guess, but know how the teacher/prof scores. I had one prof who would deduct points for wrong answers. This changes the math on guessing a lot! And I love that I guessed the term ‘no-no’ 😎 but, I disagree that one cannot/ should not cancel the x/x at the earliest opportunity. No need to drag terms around beyond when they can just ‘go away’.
X^4-X^2
Thanks. Happy to report that I got it right first time.
John,
It seems that, in the second part of the video, everything gets cancelled. But I can't see the result being 1. You crossed the whole fraction. Cansomeone help me with this,please? Thanks
Not sure if you still need help, but if so I hope this helps.
Do you understand that 4 - 5 is the same thing as -5 + 4. If so, then apply that concept to the numerator. Now the numerator is -x² + x. We want that to be exactly the same as the denominator which we can do by multiplying by -1. (We are NOT changing the values of x, just the signs of subtraction or addition.)
So we now have -1(x² - x). The denominator is really the same thing as 1(x²-x). so Cross out the terms inside the () and you are left with -1 divided by 1.
The answer is “None of the above”. This is still a function of X so it depends on what you plug in for X. The answer is -1 for most values of X and “Undefined” for x = 0 or x = 1.
They're nothing to "plug-in" as x at the end, the xs cancel. I got -1 in my head in about 10 seconds. This guy drags it out. You just multiply by -1/-1 and only process the numerator first and you end up cancelling the original numerator and denominator and are just left with the factor 1/-1 that you held back and this = -1.
Simplifying this equation involves factoring out X in both the numerator and the denominator an then dividing. In most cases X/X = 1 and can be dropped. This is not true when X happens to be 0. This example is interesting ibecause X completely disappears from the simplified function. The problem does not provide information on how to handle discontinuities so the domain of the original function must apply to the domain of the simplified result.
Ans=--1. --1(x^2--x)/(x^2--v)=--1
c) -1
Thank you
Is always worth to substitute the x in your mind before engaging in something more laborious. Here making the x simply 2 solves the sentence: x = -1
That only proves the special case for x=2. We need the answer for any possible value of x.
Although given that this is a multiple choice question, if we assume that one of the options is going to be correct then your approach is sufficient.
@@gavindeane3670 Hi Gavin you are right, I am learning everyday.
Thank You
Just multiply top and bottom by -1.
-1 ( x - x²) (x² - x) 1
____________ = _________ = _______.
-1 (x² -×) -1 (x² - x) -1 (1)
= -1
I learned a lot from your UA-cam channel and videos and I what to say thank you 📓🖋🩷🩵❤️💜💖💗💓💕💞
Got it right can't believe it Mr J ❤ 👋💪🙏🌎
This is the good stuff
I have a serious math problem 😂
I'm addicted to math
but only if x is not 0 and x is not 1.
factor the negative....great thanks for the fun
I factored. Then I put
In several values for x. They all worked out to negative one
However if x=0, this results in 0/0 and you cannot divide by zero
Also got it right, but it took a few minutes and some serious thinkin'. But before these videos, I might have been totally lost.
-1. Is the answer
-1÷1 = -1
Hi,
please excuse my bad English,
0 and 1 and -1 all fit by filling in, but calculating the solution is only -1😵💫 i m puzzled
Greetz from Germany
How are you getting the answers 0 or 1?
@@gavindeane3670Sorry für bothering, Zero + 1 are forbidden, because the denominator must n BE Zero✌️
(A+B) = - (B+A) right? I got the correct answer.
I think you over complicated the algebraic solution. Setting aside the fact that x = {0,1} yields an undefined solution, since that was not one of the multiple choices, for all other real number values of x, (1-x)/(x-1)=-1.
C -1
C = -1
-1
Suppose x=0 or 1. Then what?#!
C
1
Answer X = - 1
Nice
Solved in my head at the thumbnail in about 20 seconds, my answer is c) -1.
x/2
_1
-1 in about three seconds. Seriously? "MANY WILL GET WRONG" crap again? Can't you just teach without baiting? It's demeaning to those who are struggling. This is the reason people say, "Math is hard!" You put them down before they ever get the chance to learn. Telling them they're just part of the group who get the problem wrong only reinforces their anguish.
Be human. Stop with that "MANY WILL GET WRONG" crap. Teach the math.
Math lover.-x divided by x=-1
You are a teacher with 630k students, so you have a responsibility to pronounce math terminology correctly. It's algebraic, not algebaric, so it's pronounced al-juh-bray-ic, not al-juh-bare-ic.
Mr it's. Well said. How did this person became a math teacher. Decayed not decade
c) -1 AND a) 0. I have to disagree with the basic rule that a fraction's denominator can never be zero. I hereby declare that it CAN be zero when the numerator is also zero. So 0/0 =0. In this special case, you CAN divide by zero because nothing divided by nothing is nothing. That is intuitively correct. If the numerator is any number other than zero, the rule is true. Therefore answer a) 0 is also correct, when X = 0.
Division by zero is undefined.
@@richardhole8429 Not any more. I just defined it. It's called evolution. You learn something new every day. :)
@@terry_willis
Division by 0 is ∞ and it can be proven.
All you have to do is evaluate the following:
lim (1/x)
x>>0
Now start X at say 1
This evaluates to 1
Now set x = .00001
This evaluates to 100,000
So as x approaches 0, 1/x approaches ∞
So division by 0 has no finite answer.
don't know if you noticed, but it is asking for the value of the equation (the question mark), not for X.🙂
@@topkatz58 you are correct about the asymptote approaching infinity but it never gets to zero. Division by zero is undefined
Tricky
Not at all if you have strong math skills. I would expect one of my C students to be able to do this without multiple choice in about 30 seconds.
Letters can never equal numbers!
Please don't talk and waste in subscription matters come to point
One for the kiddies: If you take 3 smarties and divide by nothing what do you have?
STILL 3 SMARTIES! (.munch.munch. ☺️😋)
If it's -2 outside and the weather man says it is going to get warmer by one-half, what is the temperature going to be? -1
🎉. TA-DA! 🎉
c) -1
-1
C
1
_1
-1
C