Question

Solve for $x:$ $4x^2-4ax+(a^2-b^2)=0.$

Answer 1

The given quadratic equation is $4x^2-4ax+(a^2-b^2)=0$

We have $4x^2-4ax+(a+b)(a-b)=0$

$\Rightarrow 4x^2+[-2a-2a+2b-2b]x+(a-b)(a+b)=0$ [$\because$ We can write $-4a=[-2a-2a+2b-2b]$]

$\Rightarrow 4x^2+(2b-2a)x-(2a+2b)x+(a-b)(a+b)=0$

$\Rightarrow 4x^2+2(b-a)x-2(a+b)x+(a-b)(a+b)=0$

$\Rightarrow 2x[2x-(a-b\left)\right]-(a+b)[2x-(a-b\left)\right]=0$

$\Rightarrow [2x-(a-b\left)\right][2x-(a+b\left)\right]=0$

$\Rightarrow 2x-(a-b)=0,2x-(a+b)=0$

$\Rightarrow x=\frac{a+b}{2},\frac{a-b}{2}.$