Question

Find the real solutions of the equation$x^2=\frac{3}{4}x-\frac{1}{8}$ by completing square.

Short answer

The real solution of the equation is $x=\frac{1}{2}\text{and}x=\frac{1}{4}$.

Step-by-step solution

Step 1. Complete the square of given quadratic.

$$\begin{gathered} x^2=\frac{3}{4}x-\frac{1}{8}\mathrm{....}\left(\text{Given}\right) \\ {x}^2-\frac{3}{4}x=-\frac{1}{8}\mathrm{....}\left(\text{Rearranged}\right) \\ {x}^2-\frac{3}{4}x+\left(\frac{9}{64}\right)=-\frac{1}{8}+\left(\frac{9}{64}\right)\mathrm{....}\left(\text{Add}{\left(-\frac{3}{8}\right)}^2\text{\hspace{0.33em}bothsides}\right) \\ {\left(x-\frac{3}{8}\right)}^2=\frac{1}{64}\mathrm{....}\left(1\right) \end{gathered}$$

Step 2. Further simplifying (1).

$$\begin{gathered} x-\frac{3}{8}=\pm \frac{1}{8}\mathrm{....}\left(\text{Squarerootof}\left(1\right)\right) \\ x=\frac{3}{8}\pm \frac{1}{8} \\ \text{or,}x=\frac{3}{8}-\frac{1}{8},\frac{3}{8}+\frac{1}{8} \\ x=\frac{1}{4}\text{or}x=\frac{1}{2} \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

From the obtained values, the real solution of the equation is $x=\frac{1}{2}\text{and}x=\frac{1}{4}$.