Question

If $a$ and $b$ are the non-zero distinct roots of $x^2+ax+b=0$, then the least value of quadratic polynomial $x^2+ax+b$ is

• A. $1$
• B. $\frac{2}{3}$
• C. $\frac{9}{4}$
• D. $-\frac{9}{4}$

Answer 1

The correct option is D $-\frac{9}{4}$

$a,b$ are the roots of $x^2+ax+b=0$
$\Rightarrow a+b=-a\text{and}a\cdot b=b$
$\Rightarrow a=1,b=-2(\because a,b\ne 0)$
Least value of $x^2+ax+b$
$\frac{4\left(1\right)\left(b\right)-a^2}{4\left(1\right)}=-\frac{9}{4}$

Alternate Solution:
$x^2+x-2={(x+\frac{1}{2})}^2-\frac{9}{4}$
$\Rightarrow$ Least value $=-\frac{9}{4}$