Find the Roots of the Quadratic Equation by Completing the Square: x² − 4√2x + 6 = 0
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ x^2 – 4\sqrt{2}x + 6 = 0 \]Solution
\[
x^2 – 4\sqrt{2}x = -6
\]
Add the square of half the coefficient of \(x\) to both sides:
\[
x^2 – 4\sqrt{2}x + \left(\frac{4\sqrt{2}}{2}\right)^2
=
-6 + \left(2\sqrt{2}\right)^2
\]
\[
x^2 – 4\sqrt{2}x + 8 = -6 + 8
\]
\[
(x – 2\sqrt{2})^2 = 2
\]
Taking square roots:
\[
x – 2\sqrt{2} = \pm \sqrt{2}
\]
\[
x = 2\sqrt{2} \pm \sqrt{2}
\]
Therefore,
\[
x = 3\sqrt{2}
\]
or
\[
x = \sqrt{2}
\]
Answer
\[
\boxed{x = 3\sqrt{2} \quad \text{or} \quad x = \sqrt{2}}
\]