Find the Roots of x² − (√2 + 1)x + √2 = 0 by Completing the Square
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ x^2-(\sqrt2+1)x+\sqrt2=0 \]Solution
\[
x^2-(\sqrt2+1)x=-\sqrt2
\]
Adding the square of half the coefficient of \(x\) to both sides,
\[
\left(x-\frac{\sqrt2+1}{2}\right)^2
=
-\sqrt2+\frac{(\sqrt2+1)^2}{4}
\]
\[
=
-\sqrt2+\frac{3+2\sqrt2}{4}
=
\frac{3-2\sqrt2}{4}
\]
Since
\[
3-2\sqrt2=(\sqrt2-1)^2,
\]
\[
\left(x-\frac{\sqrt2+1}{2}\right)^2
=
\left(\frac{\sqrt2-1}{2}\right)^2
\]
Taking square roots,
\[
x-\frac{\sqrt2+1}{2}
=
\pm\frac{\sqrt2-1}{2}
\]
Hence,
\[
x=\sqrt2
\quad \text{or} \quad
x=1
\]
Answer
\[
\boxed{x=\sqrt2 \quad \text{or} \quad x=1}
\]