Question

Find the roots of the following quadratic equation if they exist by the method of completing the square : $\begin{array}{l}2x^2+x+4=0\end{array}$.

Answer 1

Step 1: Solve using the method of complete square

The given equation is,

$\begin{array}{l}2x^2+x+4=0\end{array}$

$\Rightarrow$$\begin{array}{l}{x}^2+\frac{x}{2}+2=0\end{array}$

$\Rightarrow$ $\begin{array}{l}{x}^2+\frac{x}{2}=–2\end{array}$

Now by adding $\begin{array}{l}(\frac{1}{4}{)}^2\end{array}$ on both sides

$\begin{array}{l}\Rightarrow x^2+\frac{x}{2}+(\frac{1}{4}{)}^2=-2+(\frac{1}{4}{)}^2\end{array}$

$\Rightarrow$ $\begin{array}{l}(x+\frac{1}{4}{)}^2=-2+\frac{1}{16}\end{array}$

$\Rightarrow$ $\begin{array}{l}(x+\frac{1}{4}{)}^2=\frac{-31}{16}\end{array}$

Step 2: Determine the Roots

Now by taking square root on both sides

$\begin{array}{l}\Rightarrow x+\frac{1}{4}=\frac{\pm \sqrt{-31}}{4}\end{array}$

$\Rightarrow$ $\begin{array}{l}x=\frac{\sqrt{-31}-1}{4},x=\frac{-\sqrt{-31}-1}{4}\end{array}$

Hence, the roots of the given equation are $\frac{\sqrt{-31}-1}{4}$ and $\frac{-\sqrt{-31}-1}{4}$.