Answer
$x=\frac{7}{4}$
Work Step by Step
$\sqrt{4\sqrt{4x+9}}=\sqrt{8x+2}$
Squaring both sides of the equation...
$4\sqrt{4x+9}=8x+2$
Squaring both sides again...
$16(4x+9)=64x^2+32x+4$
$64x+144=64x^2+32x+4$
Substracting $64x+144$ from both sides
$64x^2-32x-140=0$
Dividing both sides of an equation by 4
$16x^2-8x-35=0$
Solving for the quadratic equation using quadratic formula...
$x=\frac{8\pm \sqrt{(8)^2+4(16)(35)}}{2\times16}=\frac{8\pm48}{32}$ $=\frac{7}{4}$ or $x=-\frac{5}{4}$
We check the solutions:
$\sqrt{4\sqrt{4\cdot\frac{7}{4}+9}}=\sqrt{16}=4$
$\sqrt{8\cdot\frac{7}{4}+2}=\sqrt{16}=4$
$4=4\checkmark$
$\sqrt{4\sqrt{4\cdot\left(-\frac{5}{4}\right)+9}}=\sqrt{8}=2\sqrt 2$
$\sqrt{8\cdot\left(-\frac{5}{4}\right)+2}=\sqrt{-8}=2i\sqrt 2$
$2\sqrt 2\not=2i\sqrt 2$
The only solution is $x=\frac{7}{4}$.
