Question:
Find the roots of the following quadratic equations (if they exist) by the method of completing the square.
$x^{2}-4 a x+4 a^{2}-b^{2}=0$
Solution:
We have to find the roots of given quadratic equation by the method of completing the square. We have,
$x^{2}-4 a x+4 a^{2}-b^{2}=0$
Now shift the constant to the right hand side,
$x^{2}-4 a x=b^{2}-4 a^{2}$
Now add square of half of coefficient ofon both the sides,
$x^{2}-2(2 a) x+(2 a)^{2}=b^{2}-4 a^{2}+(2 a)^{2}$
We can now write it in the form of perfect square as,
$(x-2 a)^{2}=b^{2}$
Taking square root on both sides,
$(x-2 a)=\sqrt{b^{2}}$
So the required solution of,
$x=2 a \pm b$
$=2 a+b, 2 a-b$