Find the Roots of the Quadratic Equation by Completing the Square: √3x² + 10x + 7√3 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ \sqrt{3}x^2 + 10x + 7\sqrt{3} = 0 \]Solution
\[
\sqrt{3}x^2 + 10x = -7\sqrt{3}
\]
Divide both sides by \(\sqrt{3}\):
\[
x^2 + \frac{10}{\sqrt{3}}x = -7
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 + \frac{10}{\sqrt{3}}x + \left(\frac{5}{\sqrt{3}}\right)^2
=
-7 + \frac{25}{3}
\]
\[
\left(x+\frac{5}{\sqrt{3}}\right)^2
=
\frac{-21+25}{3}
=
\frac{4}{3}
\]
Taking square roots on both sides:
\[
x+\frac{5}{\sqrt{3}}
=
\pm\frac{2}{\sqrt{3}}
\]
\[
x
=
-\frac{5}{\sqrt{3}}
\pm
\frac{2}{\sqrt{3}}
\]
Therefore,
\[
x=-\frac{3}{\sqrt{3}}
=-\sqrt{3}
\]
or
\[
x=-\frac{7}{\sqrt{3}}
=-\frac{7\sqrt{3}}{3}
\]
Answer
\[
\boxed{x=-\sqrt{3}\quad \text{or}\quad x=-\frac{7\sqrt{3}}{3}}
\]