Question

If $2a$ and $2b$ ($\ne 0$) are the roots of the equation $x^2+ax+b=0$, then the least value of $x^2+ax+b$ ($xϵR$) occur at $x=$

• A. $\frac{-1}{16}$
• B. $\frac{1}{4}$
• C. $\frac{-1}{4}$
• D. $\frac{-1}{8}$

Answer 1

The correct option is D $\frac{-1}{8}$

$x^2+ax+b=0$

roots are $2a$ and $2b$

sum $=2(a+b)=-a$ - equation 1

product $=4ab=b$ - equation 2

from 1 and 2 equations we get

$4a=1$

$a=\frac{1}{4}$

from equation 1 we get

$2b=-3a$

$b=\frac{-3}{4}$

equation = $x^2+\frac{1}{4}x-\frac{3}{4}=0$

for a quadratic equation ${ax}^2+bx+c=0$

$x=\frac{-\left(1/4\right)}{2}=\frac{-1}{8}$