Question

$(2x+6)^2+(2x+6)^2=(6x-10)^2$

Explanation:

Step1: Combine like terms

Since the two left terms are identical, add them:
$2(2x+6)^2=(6x-10)^2$

Step2: Expand all squared terms

Use $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$:
$2(4x^2+24x+36)=36x^2-120x+100$
$8x^2+48x+72=36x^2-120x+100$

Step3: Rearrange to standard quadratic form

Move all terms to one side to set equation to 0:
$0=36x^2-120x+100-8x^2-48x-72$
$28x^2-168x+28=0$
Divide entire equation by 28 to simplify:
$x^2-6x+1=0$

Step4: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=1$, $b=-6$, $c=1$:
$x=\frac{6\pm\sqrt{(-6)^2-4(1)(1)}}{2(1)}$
$x=\frac{6\pm\sqrt{36-4}}{2}$
$x=\frac{6\pm\sqrt{32}}{2}$
$x=\frac{6\pm4\sqrt{2}}{2}$
$x=3\pm2\sqrt{2}$

Answer:

$x=3+2\sqrt{2}$ and $x=3-2\sqrt{2}$