Thanks for showing how to find the irrational solutions! 1/4 was pretty obvious, but the whole point is to use a method that gives us all the solutions.
for some reason on all of these kind of videos, there are always people who take the real root and believe it was be so simple, whereas, if they would just pay attention to the video, they might find out some thing about complex roots or at least get all of the roots in the equation instead of just the first one. it's always the same comment: why did you do all that math? I got the solution in just 2 minutes.
OK, the first solution is easy: x1 = 1/4. 4^3 = 64 => (1/4)^3 = 1/64. What we are seeking for is 3/64 so the first term has to be 4/64 = 1/16 = 1/4². Doing polynomial division by (x - 1/4) yields - x² - (3/4)x - (3/16). Using the standard formula yields x2 and x3 = (3 +/- SQRT(21))/8 My first guess was that the two other solution would be complex ones but I was wrong
Thanks for showing how to find the irrational solutions! 1/4 was pretty obvious, but the whole point is to use a method that gives us all the solutions.
for some reason on all of these kind of videos, there are always people who take the real root and believe it was be so simple, whereas, if they would just pay attention to the video, they might find out some thing about complex roots or at least get all of the roots in the equation instead of just the first one. it's always the same comment: why did you do all that math? I got the solution in just 2 minutes.
OK, the first solution is easy: x1 = 1/4. 4^3 = 64 => (1/4)^3 = 1/64. What we are seeking for is 3/64 so the first term has to be 4/64 = 1/16 = 1/4².
Doing polynomial division by (x - 1/4) yields - x² - (3/4)x - (3/16).
Using the standard formula yields x2 and x3 = (3 +/- SQRT(21))/8
My first guess was that the two other solution would be complex ones but I was wrong
👍👏👏
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A Nice Algebra Equation: x² - x³ = 3/64; x =?
Let: a = 1/4, a² = (1/4)² =1/16, a³ = (1/4)³ = 1/64; x² - x³ = 3/64 = 3a³
x³ - x² + 3a³ = 0, x³ - a³ - x² + 4a³ = (x³ - a³) - [x² - 4(1/64)] = (x³ - a³) - (x² - 1/16) = 0 (x³ - a³) - (x² - a²) = (x - a)(x² + ax + a²) - (x - a)(x + a) = 0
(x - a)(x² + ax + a² - x - a) = 0, x - a = 0 or x² + ax + a² - x - a = 0
x = a = 1/4 or x² + x/4 + 1/16 - x - 1/4 = x² - (3/4)x - 3/16 = 0, x² - (3/4)x = 3/16
x² - (3/4)x + (3/8)² = (x - 3/8)² = 3/16 + (3/8)² = 21/64 = [(√21)/8]²; x - 3/8 = ± (√21)/8
x = 3/8 ± (√21)/8 = (3 ± 3√3)/8
Answer check:
x = 1/4: x² - x³ = (1/4)² - (1/4)³ = 1/16 - 1/64 = 3/64; Confirmed
x = (3 ± 3√3)/8: x² - (3/4)x - 3/16 = 0; x² = (3/4)x + 3/16, x³ = (3/4)x² + (3/16)x
x² - x³ = [(3/4)x + 3/16] - [(3/4)x² + (3/16)x] = 3/16 + (9/16)x - (3/4)x²
= 3/16 + (9/16)x - (3/4)[(3/4)x + 3/16] = 3/16 - 9/64 = 3/64; Confirmed
Final answer:
x = 1/4, x = (3 + 3√3)/8 or (3 - 3√3)/8
you are brilliant!
Thanks 🙏
0.25
8.
1/4
x=1/4
3/64 is actually (1/4)^2 - (1/4)^3
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