Question

Find the real solutions of the equation$x^2-6x-11=0$ by completing square.

Short answer

The real solution of the equation is $x=3-2\sqrt{5}\text{and}x=3+2\sqrt{5}$.

Step-by-step solution

Step 1. Complete the square of given quadratic.

$$\begin{gathered} x^2-6x-11=0....\left(\text{Given}\right) \\ {x}^2-6x=11....\left(\text{Add}11\text{bothsides}\right) \\ {x}^2-6x+\left(9\right)=11+\left(9\right)\mathrm{....}\left(\text{Add}{\left(-\frac{6}{2}\right)}^2\text{\hspace{0.33em}bothsides}\right) \\ {\left(x-3\right)}^2=20....\left(1\right) \end{gathered}$$

Step 2. Further simplifying (1).

$$\begin{gathered} \left(x-3\right)=\pm \sqrt{20}\mathrm{....}\left(\text{Squarerootof}\left(1\right)\right) \\ \left(x-3\right)=\pm 2\sqrt{5} \\ \text{or,}x=3\pm 2\sqrt{5} \\ x=3-2\sqrt{5}\text{or}x=3+2\sqrt{5} \end{gathered}$$

Step 3. Description of steps.

From the obtained values, the real solution of the equation is $x=3-2\sqrt{5}\text{and}x=3+2\sqrt{5}$.