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

Answer 1

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

Let the quadratic equation is
$f\left(x\right)=x^2+\frac{b}{a}x+\frac{c}{a}$

As, $-1$ and $\beta$ are the roots of quadratic equation
$\therefore -1+\beta =-\frac{b}{a}\Rightarrow \frac{b}{a}=1-\beta$
and $(-1)\beta =\frac{c}{a}\Rightarrow \frac{c}{a}=-\beta$

So, $f\left(x\right)=x^2+(1-\beta )x-\beta$

Now, $f\left(1\right)+f\left(2\right)=0$

$\Rightarrow (1+(1-\beta )-\beta )+(4+(1-\beta )2-\beta )=0$

$\Rightarrow \beta =\frac{8}{5}$