Find the Roots of the Quadratic Equation by Completing the Square: 2x² + x − 4 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ 2x^2 + x – 4 = 0 \]Solution
\[
2x^2 + x = 4
\]
Divide both sides by 2:
\[
x^2 + \frac{x}{2} = 2
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 + \frac{x}{2} + \left(\frac{1}{4}\right)^2
=
2 + \frac{1}{16}
\]
\[
\left(x+\frac{1}{4}\right)^2
=
\frac{33}{16}
\]
Taking square roots on both sides:
\[
x+\frac{1}{4}
=
\pm\frac{\sqrt{33}}{4}
\]
\[
x
=
-\frac{1}{4}
\pm
\frac{\sqrt{33}}{4}
\]
Hence,
\[
x=\frac{-1+\sqrt{33}}{4}
\]
or
\[
x=\frac{-1-\sqrt{33}}{4}
\]
Answer
\[
\boxed{
x=\frac{-1+\sqrt{33}}{4}
\quad \text{or} \quad
x=\frac{-1-\sqrt{33}}{4}
}
\]