Find the Roots of the Quadratic Equation by Completing the Square: 3x² + 11x + 10 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ 3x^2 + 11x + 10 = 0 \]Solution
\[
3x^2 + 11x = -10
\]
Divide both sides by 3:
\[
x^2 + \frac{11}{3}x = -\frac{10}{3}
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 + \frac{11}{3}x + \left(\frac{11}{6}\right)^2
=
-\frac{10}{3} + \frac{121}{36}
\]
\[
\left(x+\frac{11}{6}\right)^2
=
-\frac{120}{36}+\frac{121}{36}
=
\frac{1}{36}
\]
Taking square roots:
\[
x+\frac{11}{6}
=
\pm\frac{1}{6}
\]
\[
x
=
-\frac{11}{6}\pm\frac{1}{6}
\]
Therefore,
\[
x=-\frac{10}{6}=-\frac{5}{3}
\]
or
\[
x=-\frac{12}{6}=-2
\]
Answer
\[
\boxed{x=-\frac{5}{3} \quad \text{or} \quad x=-2}
\]