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

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

Solve for A

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

Solution Steps

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

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

Consider \left(x+2\right)\left(x-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Subtract 4 from 4 to get 0.

$$A=2x^{2}-4x$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{2A+4}}{2}+1$$

$$x=\frac{\sqrt{2A+4}}{2}+1$$

Solve for x

$$x=-\frac{\sqrt{2A+4}}{2}+1$$

$$x=\frac{\sqrt{2A+4}}{2}+1\text{, }A\geq -2$$