What are the solutions to the equation x^3-6x^2+15x-7=(x+a)^3+bx+c ?

The solutions to the equation x^3-6x^2+15x-7=(x+a)^3+bx+c are

Find the zeros of x^3-6x^2+15x-7=(x+a)^3+bx+c

The zeros of x^3-6x^2+15x-7=(x+a)^3+bx+c are

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x^{3} - 6 x^{2} + 15 x - 7 = (x + a)^{3} + b x + c

x = \frac{3 a ^{2} + b - 15 + - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}, x = \frac{3 a ^{2} + b - 15 - - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}; a \neq = - 2

Solution steps

x^{3} - 6 x^{2} + 15 x - 7 = (x + a)^{3} + b x + c

Expand

(x + a)^{3} + b x + c : x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c

x^{3} - 6 x^{2} + 15 x - 7 = x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c

Subtract

x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c

from both sides

x^{3} - 6 x^{2} + 15 x - 7 - (x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c) = x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c - (x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3} + b x + c)

Simplify

- (3 a + 6) x^{2} + (- 3 a^{2} - b + 15) x - a^{3} - c - 7 = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( - 3 a ^{2} - b + 15 ) \pm ( - 3 a ^{2} - b + 15 ) ^{2} - 4 ( - 3 a - 6 ) ( - a ^{3} - c - 7 )}{2 ( - 3 a - 6 )}

Simplify

(- 3 a^{2} - b + 15)^{2} - 4 (- 3 a - 6) (- a^{3} - c - 7) : - 3 a^{4} - 24 a^{3} + 6 a^{2} b - 90 a^{2} - 12 a c - 84 a + b^{2} + 57 - 30 b - 24 c

x_{1, 2} = \frac{- ( - 3 a ^{2} - b + 15 ) \pm - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}; a \neq = - 2

Separate the solutions

x_{1} = \frac{- ( - 3 a ^{2} - b + 15 ) + - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}, x_{2} = \frac{- ( - 3 a ^{2} - b + 15 ) - - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}

x = \frac{- ( - 3 a ^{2} - b + 15 ) + - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )} : \frac{3 a ^{2} + b - 15 + - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}

x = \frac{- ( - 3 a ^{2} - b + 15 ) - - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )} : \frac{3 a ^{2} + b - 15 - - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}

The solutions to the quadratic equation are:

x = \frac{3 a ^{2} + b - 15 + - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}, x = \frac{3 a ^{2} + b - 15 - - 3 a ^{4} - 24 a ^{3} + 6 a ^{2} b - 90 a ^{2} - 12 a c - 84 a + b ^{2} + 57 - 30 b - 24 c}{2 ( - 3 a - 6 )}; a \neq = - 2

0 = - x^{2} + 3 x + 4

- 3 x^{2} - 12 x - 8 = 0

- x^{2} - \frac{2}{5} x = 1

9 x^{2} - 12 x = - 4

5 x^{2} - 9 x - 18 = 0

What are the solutions to the equation x^3-6x^2+15x-7=(x+a)^3+bx+c ?
The solutions to the equation x^3-6x^2+15x-7=(x+a)^3+bx+c are
Find the zeros of x^3-6x^2+15x-7=(x+a)^3+bx+c
The zeros of x^3-6x^2+15x-7=(x+a)^3+bx+c are