Solve A=(2x-3)^2-(x+1)^2

$$A = ( 2 x - 3 ) ^ { 2 } - ( x + 1 ) ^ { 2 }$$

Solve for A

$$A=\left(x-4\right)\left(3x-2\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.

$$A=4x^{2}-12x+9-\left(x+1\right)^{2}$$

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

$$A=4x^{2}-12x+9-\left(x^{2}+2x+1\right)$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$A=4x^{2}-12x+9-x^{2}-2x-1$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$A=3x^{2}-12x+9-2x-1$$

Combine -12x and -2x to get -14x.

$$A=3x^{2}-14x+9-1$$

Subtract 1 from 9 to get 8.

$$A=3x^{2}-14x+8$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{3A+25}+7}{3}$$

$$x=\frac{\sqrt{3A+25}+7}{3}$$

Solve for x

$$x=\frac{-\sqrt{3A+25}+7}{3}$$

$$x=\frac{\sqrt{3A+25}+7}{3}\text{, }A\geq -\frac{25}{3}$$