Question

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

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

Answer 1

Answer: (C)

$-\frac{9}{4}$

If $a,b$ are the roots of the equation $x^2+ax+b=0$,

Then $a+b=-a\Rightarrow b=-2a$ and $ab=b\Rightarrow a=1$

Hence, $b=-2$

So, the quadratic equation $=x^2+x-2$

The minimum value of the function exists at $x=-\frac{1}{2}$

The least value is ${(-\frac{1}{2})}^2-\frac{1}{2}-2=-\frac{9}{4}$