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
\[
2x^2+x=-4
\]
Dividing both sides by 2,
\[
x^2+\frac{x}{2}=-2
\]
Adding the square of half the coefficient of \(x\) to both sides,
\[
x^2+\frac{x}{2}+\left(\frac{1}{4}\right)^2
=
-2+\frac{1}{16}
\]
\[
\left(x+\frac{1}{4}\right)^2
=
-\frac{31}{16}
\]
Since
\[
\left(x+\frac{1}{4}\right)^2 \ge 0
\]
for every real value of \(x\), but
\[
-\frac{31}{16}<0,
\]
the equation cannot be satisfied by any real number.
Answer
\[
\boxed{\text{The quadratic equation } 2x^2+x+4=0 \text{ has no real roots.}}
\]