Question

Solve the following quadratic equation:
$9x^2-9(a+b)x+(2a62+5ab+2b^2)=0$

Answer 1

$9x^2-9(a+b)x+(2a^2+5ab+2b^2)=0$

Comparing given equation with the standard form $A{x}^2+Bx+C=0$, we get

$A=9,B=-9(a+b),C=2a^2+5ab+2b^2$

Using quadratic formula,

$x=\frac{-B\pm \sqrt{B^2-4AC}}{2A}$

So,

$x=\frac{9(a+b)\pm \sqrt{81(a+b{)}^2-36(2a^2+5ab+2b^2)}}{18}$

$=\frac{9(a+b)\pm 3\sqrt{9a^2+9b^2+18ab-8a^2-20ab-8b^2}}{18}$

$=\frac{9(a+b)\pm 3\sqrt{a^2+b^2-2ab}}{18}$

$=\frac{9(a+b)\pm 3\sqrt{(a-b{)}^2}}{18}$

$=\frac{9(a+b)\pm 3(a-b)}{18}$

$\Rightarrow x=\frac{9a+9b+3a-3b}{18}orx=\frac{9a+9b-3a+3b}{18}\Rightarrow x=\frac{2a+b}{3}Orx=\frac{a+b}{3}$