Question

Solve the following quadratic equations by factorization:
(x-a/x-b)+(x-b/x-a)=(a/b)+(b/a)

Answer 1

Therefore the solution of the quadratic equation is

x= 0,(a+b).

Step-by-step explanation:

Quadratic equation: The power of variable is maximum.

Given quadratic equation is,

$\frac{x-a}{x-b}+\frac{x-b}{x-a}=\frac{a}{b}+\frac{b}{a}$

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

$\Rightarrow \frac{x^2-2ax+a^2+x^2-2bx+b^2}{x^2-ax-bx+ab}=\frac{a^2+b^2}{ab}$

$\Rightarrow ab(2x^2-2ax-2bx+a^2+b^2)=(a^2+b^2)(x^2-ax-bx+ab)$

$\Rightarrow 2abx^2-2a^2bx-2ab^2x+a^3b+ab^3=x^2a^2-a^3x-a^2bx+a^3b+x^2b^2-ab^2x-b^3x+ab^3$

$\Rightarrow x^2(a^2+b^2-2ab) +x(2a^2b+2ab^2-a^2b-ab^2-a^3-b^3)=0$

$\Rightarrow x^2(a-b)^2+x(a^2b+ab^2-a^3-b^3)=0$

$\Rightarrow x[x(a-b)^2+a^2b +ab^2-a^3-b^3]=0$

$\Rightarrow x= 0, \frac{a^3+b^3-a^2b-ab^2}{(a-b)^2}$

$\Rightarrow x=0, \frac{a^3-a^2b+b^3-ab^2}{(a-b)^2}$

$\Rightarrow x= 0, \frac{a^2(a-b)-b^2(a-b)}{(a-b)^2}$

$\Rightarrow x=0, \frac{(a^2-b^2)(a-b)}{(a-b)^2}$

$\Rightarrow x= 0, \frac{(a+b)(a-b)^2}{(a-b)^2}$

⇒ x=0, (a+b)

Therefore the solution of the quadratic equation is

x= 0,(a+b)