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

For the following equation, find the value of x.

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}