Chapter SH, Problem 4.15EP

To determine

Find the solution of the given equation using the ‘completing the square’ method.

Answer to Problem 4.15EP

The solutions of the given equation are $x=-1+\sqrt{\frac{7}{3}}i\text{and}x=-1-\sqrt{\frac{7}{3}}i$ .

Explanation of Solution

Given:

The given equationis $3x^2+6x+10=0$ .

Calculation:

  $3x^2+6x+10=0$

Subtract $10$ from both sides.

  $\begin{array}{l}3x^2+6x+10-10=0-10\\ \Rightarrow 3x^2+6x=-10\end{array}$

Divide both sides by $3$ .

  $\begin{array}{l}\Rightarrow \frac{3}{3}{x}^2+\frac{6}{3}x=-\frac{10}{3}\\ \Rightarrow x^2+2x=-\frac{10}{3}\end{array}$

Apply identity $x^2\pm 2xy+y^2={\left(x\pm y\right)}^2$ in the left side.

  $\begin{array}{l}\Rightarrow x^2+2x+{\left(\frac{2}{2}\right)}^2=-\frac{1}{5}+{\left(\frac{2}{2}\right)}^2\\ \Rightarrow x^2+2x+1=-\frac{10}{3}+1\\ \Rightarrow x^2+2x+1=-\frac{7}{3}\\ \Rightarrow {\left(x+1\right)}^2=-\frac{7}{3}\\ \Rightarrow x+1=\pm \sqrt{-\frac{7}{3}}\\ i=\sqrt{-1}\\ \Rightarrow x+1=\pm \sqrt{\frac{7}{3}}i\\ \Rightarrow x+1=\sqrt{\frac{7}{3}}i\text{and}\Rightarrow x+1=-\sqrt{\frac{7}{3}}i\\ \Rightarrow x+1-1=-1+\sqrt{\frac{7}{3}}i\text{and}\Rightarrow x+1-1=-1-\sqrt{\frac{7}{3}}i\\ \Rightarrow x=-1+\sqrt{\frac{7}{3}}i\text{and}\Rightarrow x=-1-\sqrt{\frac{7}{3}}i\end{array}$

Hence thesolutions of the given equation are $x=-1+\sqrt{\frac{7}{3}}i\text{and}x=-1-\sqrt{\frac{7}{3}}i$ .