What are the solutions to the equation 2x^3-8x^2-24x=ax(x+b)(x+c) ?

The solutions to the equation 2x^3-8x^2-24x=ax(x+b)(x+c) are

Find the zeros of 2x^3-8x^2-24x=ax(x+b)(x+c)

The zeros of 2x^3-8x^2-24x=ax(x+b)(x+c) are

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2 x^{3} - 8 x^{2} - 24 x = a x (x + b) (x + c)

x = 0, x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

Solution steps

2 x^{3} - 8 x^{2} - 24 x = a x (x + b) (x + c)

Move

a x (x + b) (x + c)

to the left side

2 x^{3} - 8 x^{2} - 24 x - a x (x + b) (x + c) = 0

Factor

2 x^{3} - 8 x^{2} - 24 x - a x (x + b) (x + c) : x (2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24)

x (2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x = 0 or 2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24 = 0

Solve

2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24 = 0 : x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

The solutions are

x = 0, x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

- 2 x^{3} = 32 x

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

27 x^{3} - 1 = 0

4 x^{2} - 9 = (2 x^{3}) (2 x - 3)

4 x^{4} - 64 x^{2} = 0

What are the solutions to the equation 2x^3-8x^2-24x=ax(x+b)(x+c) ?
The solutions to the equation 2x^3-8x^2-24x=ax(x+b)(x+c) are
Find the zeros of 2x^3-8x^2-24x=ax(x+b)(x+c)
The zeros of 2x^3-8x^2-24x=ax(x+b)(x+c) are