Question

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

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

Answer 1

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

As given $a$ and $b$ are the roots of the equation $x^2+ax+b=0$
$\Rightarrow$ sum of roots, $a+b=-a$
$\Rightarrow$ $b=-2a....\left(1\right)$
and products of roots $ab=b$

$\Rightarrow$ $ab-b=0$
$\Rightarrow$ $b(a-1)=0$
if $b=0$, then $a=0$
if $b\ne 0$, then $a=1$ and $b=-2$

So, the expression will be
$f\left(x\right)=x^2+x-2=x^2+2\frac{1}{2}x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2-2$

$\Rightarrow f\left(x\right)={(x+\frac{1}{2})}^2-\frac{9}{4}$

So, $f\left(x\right)$ will be minimum, if ${(x+\frac{1}{2})}^2=0$

i.e., when $x=-\frac{1}{2}$

Minimum value of function $=-\frac{9}{4}$.