Solve A=(x-2)^2-(2x+5)^2

$$A = ( x - 2 ) ^ { 2 } - ( 2 x + 5 ) ^ { 2 }$$

Solve for A

$$A=-3\left(x+1\right)\left(x+7\right)$$

Solution Steps

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

$$A=x^{2}-4x+4-\left(2x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+5\right)^{2}.

$$A=x^{2}-4x+4-\left(4x^{2}+20x+25\right)$$

To find the opposite of 4x^{2}+20x+25, find the opposite of each term.

$$A=x^{2}-4x+4-4x^{2}-20x-25$$

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

$$A=-3x^{2}-4x+4-20x-25$$

Combine -4x and -20x to get -24x.

$$A=-3x^{2}-24x+4-25$$

Subtract 25 from 4 to get -21.

$$A=-3x^{2}-24x-21$$

Solve for x (complex solution)

$$x=\frac{\sqrt{81-3A}}{3}-4$$

$$x=-\frac{\sqrt{81-3A}}{3}-4$$

Solve for x

$$x=\frac{\sqrt{81-3A}}{3}-4$$

$$x=-\frac{\sqrt{81-3A}}{3}-4\text{, }A\leq 27$$