Question

Suppose $a, b \in R , a \neq 0$ and $2 a+b \neq 0$. A root of the equation $(a+b)(a x+b)(a-b x)=\left(a^2 x-b^2\right)(a+b x)$ is

• A. $\frac{a+2 b}{2 a+b}$
• B. $\frac{2 a+b}{a+2 b}$
• C. $-\frac{a-2 b}{a+2 b}$
• D. $-\frac{a+2 b}{2 a+b}$

Answer 1

Answer: (D)

We can write the quation as
${\left[a^2 b+(a+b) a b\right] x^2+\left[\left(a^3-b^3\right)-(a+b)\left(a^2-b^2\right)\right] x} -a b^2-a b(a+b)=0 $
$\Rightarrow(2 a+b) x^2-(a-b) x-(a+2 b)=0$
Since the sum of coefficients is 0 one of the roots is 1 and the other root is $-\frac{a+2 b}{2 a+b}$.