A Nice Cubic Polynomial Equation
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- Опубликовано: 7 фев 2025
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x^3-8x=8
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x^3 - 8x = 8
x^3 - 4x - 4x - 8 = 0
x(x^2 - 4) - 4(x + 2) = 0
x(x+2)(x-2) - 4(x+2) = 0
(x+2)(x^2-2x-4) = 0
so, x = -2 or
x^2 - 2x - 4 = 0
x = (2 +- sqrt(4 - 4(-4)))/2 = (2 +- sqrt(20))/2 = 1 +- sqrt(5)
finally, x = -2, 1+sqrt(5),1-sqrt(5)
Nice!
4:54 Sometimes I replace x by (-x) in the equation to find negative solutions:
From x^3 - 8x - 8 = 0, we get to -x^3 + 8x - 8 = 0 or x^3 + 8 = 8x.
Then for x = 1 we get 1 + 8 = 9 = 8*1 false, but for x = 2 we get 2^3 + 8 = 8 + 8 = 16 = 8*2 correct.
So x = 2 is a solution to the modified equation and x = -2 is a solution to the original equation.
Good thinking!
It's not really depressed, it's seasonal affective.
Х=(4+4√(1-32/27))¹/³+(..-..)..=2√(8/ 3)cos(1/3arccos√(27/32)+2/3πn)
it took me a while to figure out the jump between lines 3 & 4 of your 3rd method.
You mean the sum of two cubes and factoring out the common factor?
exactly. I never bothered to memorize formulas for the sum of two squares or cubes. I figure in math, the less I have to memorize the better.
synthetic division
x = -2
I used the second method.
Achei mais fácil completamento de quadrados: x(x²-8)=8
X.(x²-4-4)=8
X.(x²-4)-4x-8=0
X.(x-2).(x+2)-4(x+2)=0
(x+2)[x.(x-2)-4]=0
(x+2)[x²-2x-4]=0
Logo, (x+2)=0, x=-2 e x²-2x-4=0
Resolvendo por bhaskara.