Question

Find the root of the following quadratic equation (if they exist) by the method of completing the square.
$\sqrt{3}{x}^2+10x+7\sqrt{3}=0$

Answer 1

Consider the given equation.

$\sqrt{3}{x}^2+10x+7\sqrt{3}=0$

$x^2+\frac{10}{3}x+7=0$

$x^2+2\times \frac{1}{2}\times \frac{10}{\sqrt{3}}x+7=0$

$x^2+2\times \frac{5}{\sqrt{3}}\times x+{\left(\frac{5}{\sqrt{3}}\right)}^2-{\left(\frac{5}{\sqrt{3}}\right)}^2+7=0$

${(x+\frac{5}{\sqrt{3}})}^2=\frac{25-21}{3}$

${(x+\frac{5}{\sqrt{3}})}^2=\frac{4}{3}$

$x+\frac{5}{\sqrt{3}}=\pm \frac{2}{\sqrt{3}}$

$x=-\sqrt{3},-\frac{7}{\sqrt{3}}$

Hence, this is the solution of the given equation by completing the squares.