# Equations Involving One Variable| Algebra| ENGG10 LE3| Problem 3

#### $5-x=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$

SOLUTION:

Square both sides

$\left(5-x\right)^2=x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$

But $5-x=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$. So, we can write the equation as

$\left(5-x\right)^2=x+\left(5-x\right)$

$25-10x+x^2=x+5-x$

$x^2-10x+25-5=0$

$x^2-10x+20=0$

Use the quadratic formula to solve for x

$x=\frac{-\left(-10\right)\pm \sqrt{\left(-10\right)^2-4\left(1\right)\left(20\right)}}{2\left(1\right)}$

$x=\frac{10\pm \sqrt{100-80}}{2}$

$x=\frac{10\pm \sqrt{20}}{2}$

$x=\frac{10\pm 2\sqrt{5}}{2}$

$x=5\pm \sqrt{5}$