Question

Solve the equation $x^2+\frac{1}{3}x-\frac{35}{36}=0$ by completing the squares.

Short answer

The solution for the given equation is $x=-\frac{2}{3}\pm \frac{\sqrt{17}}{2\sqrt{3}}$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is $x^2+\frac{1}{3}x-\frac{35}{36}=0$.

Step 2. Apply square completion method.

The given equation $x^2+\frac{1}{3}x-\frac{35}{36}=0$ with the help of completing the square method is solved as:

$\begin{array}{c}{x}^2+\frac{1}{3}x-\frac{35}{36}=0....\left(\text{Given}\right)\\ x^2+\frac{1}{3}x=\frac{35}{36}\mathrm{....}\left(\text{Rewrite}\right)\\ x^2+\frac{1}{3}x+\frac{4}{9}=\frac{35}{36}+\frac{4}{9}\mathrm{....}\left(\begin{array}{l}\text{Add}{\left(\frac{2}{3}\right)}^2\\ \text{bothsides}\end{array}\right)\\ {\left(x+\frac{2}{3}\right)}^2=\frac{17}{12}\\ x+\frac{2}{3}=\pm \frac{\sqrt{17}}{2\sqrt{3}}\\ x=-\frac{2}{3}\pm \frac{\sqrt{17}}{2\sqrt{3}}\end{array}$

Step 3. Conclusion.

The solution for the given equation is $x=-\frac{2}{3}\pm \frac{\sqrt{17}}{2\sqrt{3}}$.