Question

Solve each of the following quadratic equations:
$\frac{a}{(x-b)}+\frac{b}{(x-a)}=2,x\ne b,a$

Answer 1

$\frac{a}{(x-b)}+\frac{b}{(x-a)}=2$

$\frac{a(x-a)+b(x-b)}{(x-a)(x-b)}=2$

⇒$ax–a^2+bx–b^2=2x^2–2ax–2bx+2ab$

⇒ $2x^2–2ax–ax–2bx–bx+a^2+b^2+2ab=0$

⇒$2x^2–3x(a+b)+(a+b{)}^2=0$

⇒$2x^2–2x(a+b)–x(a+b)+(a+b{)}^2=0$

⇒$2x(x–(a+b\left)\right)–(a+b)(x–(a+b\left)\right)=0$

⇒ $(2x–(a+b\left)\right)(x–(a+b\left)\right)=0$

⇒$x=\frac{a+b}{2}orx=a+b$