[tex]for\ a\ quadratic\ equation\ ax^{2}+bx+c=0,\ solutions\ are\\ \\ x= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\ \\thus\ for\ 4 x^{2} -4ax+( a^{2} - b^{2} ),\ solution\ is\\ \\ x=\frac{4a \pm \sqrt{(4a)^2-4 \times 4 \times (a^2-b^2)}}{2 \times 4}\\ \\ \Rightarrow x=\frac{4a \pm \sqrt{16a^2-16a^2+16b^2}}{8}\\ \\ \Rightarrow x=\frac{4a \pm \sqrt{16b^2}}{8}\\ \\ \Rightarrow x=\frac{4a \pm 4b}{8}\\ \\ \Rightarrow x=\frac{a \pm b}{2}\\ \\ \Rightarrow x=\frac{a+b}{2}\ and\ \frac{a-b}{2} [/tex]
