Solve D=16(x-1)^2-8(x-1)(x+5)

$$D = 16 ( x - 1 ) ^ { 2 } - 8 ( x - 1 ) ( x + 5 )$$

Solve for D

$$D=8\left(x-7\right)\left(x-1\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$D=16\left(x^{2}-2x+1\right)-8\left(x-1\right)\left(x+5\right)$$

Use the distributive property to multiply 16 by x^{2}-2x+1.

$$D=16x^{2}-32x+16-8\left(x-1\right)\left(x+5\right)$$

Use the distributive property to multiply -8 by x-1.

$$D=16x^{2}-32x+16+\left(-8x+8\right)\left(x+5\right)$$

Use the distributive property to multiply -8x+8 by x+5 and combine like terms.

$$D=16x^{2}-32x+16-8x^{2}-32x+40$$

Combine 16x^{2} and -8x^{2} to get 8x^{2}.

$$D=8x^{2}-32x+16-32x+40$$

Combine -32x and -32x to get -64x.

$$D=8x^{2}-64x+16+40$$

Add 16 and 40 to get 56.

$$D=8x^{2}-64x+56$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{2D+144}}{4}+4$$

$$x=\frac{\sqrt{2D+144}}{4}+4$$

Solve for x

$$x=-\frac{\sqrt{2D+144}}{4}+4$$

$$x=\frac{\sqrt{2D+144}}{4}+4\text{, }D\geq -72$$