Find the Roots of the Quadratic Equation by Completing the Square: x² − 8x + 18 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ x^2 – 8x + 18 = 0 \]Solution
\[
x^2 – 8x = -18
\]
Adding the square of half the coefficient of \(x\) to both sides,
\[
x^2 – 8x + \left(\frac{8}{2}\right)^2
=
-18 + 16
\]
\[
x^2 – 8x + 16
=
-2
\]
\[
(x-4)^2=-2
\]
Since
\[
(x-4)^2 \ge 0
\]
for every real value of \(x\), but
\[
-2<0,
\]
the equation cannot be satisfied by any real number.
Answer
\[
\boxed{\text{The quadratic equation } x^2-8x+18=0 \text{ has no real roots.}}
\]