Find the Roots of the Quadratic Equation by Completing the Square: 4x² + 4√3x + 3 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ 4x^2 + 4\sqrt{3}x + 3 = 0 \]Solution
\[
4x^2 + 4\sqrt{3}x = -3
\]
Divide both sides by 4:
\[
x^2 + \sqrt{3}x = -\frac{3}{4}
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 + \sqrt{3}x + \left(\frac{\sqrt{3}}{2}\right)^2
=
-\frac{3}{4} + \frac{3}{4}
\]
\[
\left(x+\frac{\sqrt{3}}{2}\right)^2 = 0
\]
Taking square roots on both sides:
\[
x+\frac{\sqrt{3}}{2}=0
\]
\[
x=-\frac{\sqrt{3}}{2}
\]
Thus both roots are equal.
Answer
\[
\boxed{x=-\frac{\sqrt{3}}{2}}
\]
Hence, the quadratic equation has equal roots.