Find the Roots of x² − 4ax + 4a² − b² = 0 by Completing the Square
Question
Find the roots of the quadratic equation by the method of completing the square:
\[ x^2-4ax+4a^2-b^2=0 \]Solution
\[
x^2-4ax=b^2-4a^2
\]
Adding the square of half the coefficient of \(x\) to both sides,
\[
x^2-4ax+\left(\frac{4a}{2}\right)^2
=
b^2-4a^2+4a^2
\]
\[
(x-2a)^2=b^2
\]
Taking square roots,
\[
x-2a=\pm b
\]
Hence,
\[
x=2a+b
\]
or
\[
x=2a-b
\]
Answer
\[
\boxed{x=2a+b \quad \text{or} \quad x=2a-b}
\]