What are the solutions to the equation x^3+2x^2-9x+18=(x-a)(x-b)(x+c) ?

The solutions to the equation x^3+2x^2-9x+18=(x-a)(x-b)(x+c) are

Find the zeros of x^3+2x^2-9x+18=(x-a)(x-b)(x+c)

The zeros of x^3+2x^2-9x+18=(x-a)(x-b)(x+c) are

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x^{3} + 2 x^{2} - 9 x + 18 = (x - a) (x - b) (x + c)

x = \frac{ab - b c - a c + 9 + a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}, x = \frac{ab - b c - a c + 9 - a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}; a + b - c + 2 \neq = 0

Solution steps

x^{3} + 2 x^{2} - 9 x + 18 = (x - a) (x - b) (x + c)

Expand

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

x^{3} + 2 x^{2} - 9 x + 18 = x^{3} + c x^{2} - b x^{2} - b c x - a x^{2} - a c x + ab x + ab c

Subtract

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

from both sides

x^{3} + 2 x^{2} - 9 x + 18 - (x^{3} + c x^{2} - b x^{2} - b c x - a x^{2} - a c x + ab x + ab c) = x^{3} + c x^{2} - b x^{2} - b c x - a x^{2} - a c x + ab x + ab c - (x^{3} + c x^{2} - b x^{2} - b c x - a x^{2} - a c x + ab x + ab c)

Simplify

(a + b - c + 2) x^{2} + (- ab + b c + a c - 9) x - ab c + 18 = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( - ab + b c + a c - 9 ) \pm ( - ab + b c + a c - 9 ) ^{2} - 4 ( a + b - c + 2 ) ( - ab c + 18 )}{2 ( a + b - c + 2 )}

Simplify

(- ab + b c + a c - 9)^{2} - 4 (a + b - c + 2) (- ab c + 18) : a^{2} b^{2} + a^{2} c^{2} + 2 a^{2} b c + 18 ab + 2 a b^{2} c + 8 ab c - 2 ab c^{2} - 18 a c - 72 a + b^{2} c^{2} + 72 c - 72 b - 18 b c - 63

x_{1, 2} = \frac{- ( - ab + b c + a c - 9 ) \pm a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}; a + b - c + 2 \neq = 0

Separate the solutions

x_{1} = \frac{- ( - ab + b c + a c - 9 ) + a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}, x_{2} = \frac{- ( - ab + b c + a c - 9 ) - a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}

x = \frac{- ( - ab + b c + a c - 9 ) + a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )} : \frac{ab - b c - a c + 9 + a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}

x = \frac{- ( - ab + b c + a c - 9 ) - a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )} : \frac{ab - b c - a c + 9 - a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}

The solutions to the quadratic equation are:

x = \frac{ab - b c - a c + 9 + a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}, x = \frac{ab - b c - a c + 9 - a ^{2} b ^{2} + a ^{2} c ^{2} + 2 a ^{2} b c + 18 ab + 2 a b ^{2} c + 8 ab c - 2 ab c ^{2} - 18 a c - 72 a + b ^{2} c ^{2} + 72 c - 72 b - 18 b c - 63}{2 ( a + b - c + 2 )}; a + b - c + 2 \neq = 0

4 x^{2} - 4 a^{2} x + a^{4} - b^{4} = 0

6 = \frac{n ( n - 1 )}{2}

3^{3} (3 x)^{2} = 3^{5}

c, c^{2} = h^{2} - c^{2}

81 y^{2} - k = (\frac{9 y + 2}{3}) (\frac{9 y - 2}{3})

What are the solutions to the equation x^3+2x^2-9x+18=(x-a)(x-b)(x+c) ?
The solutions to the equation x^3+2x^2-9x+18=(x-a)(x-b)(x+c) are
Find the zeros of x^3+2x^2-9x+18=(x-a)(x-b)(x+c)
The zeros of x^3+2x^2-9x+18=(x-a)(x-b)(x+c) are