Question

If $f\left(x\right)$ is a quadratic expression such that $f\left(1\right)+f\left(2\right)=0$, and $-1$ is a root of $f\left(x\right)=0$, then the other root of $f\left(x\right)=0$ is :

• A. $-\frac{5}{8}$
• B. $-\frac{8}{5}$
• C. $\frac{5}{8}$
• D. $\frac{8}{5}$

Answer 1

The correct option is D $\frac{8}{5}$

Let $\alpha$ and $\beta =-1$ be the roots of the polynomial, then we have $f\left(x\right)=x^2+(1-\alpha )x-\alpha$.

$f\left(1\right)=2-2\alpha$ .......$i$

$f\left(2\right)=6-3\alpha$ ........$ii$

$f\left(1\right)+f\left(2\right)=0\Rightarrow 2-2\alpha +6-3\alpha =0\Rightarrow \alpha =\frac{8}{5}$

So the other root is $\frac{8}{5}$

So the correct answer is option D.