Question

If $a,b,c\in R$ and $a+b+c=0$, then the quadratic equation $4a{x}^2+3bx+2c=0$ has

• A. one positive and one negative
• B. imaginary roots
• C. real roots
• D. none of these

Answer 1

The correct option is C real roots

$4a{x}^2+3bx+2c$

Discriminant, $\Delta =(3b{)}^2-4(4a\left)\right(2c)$

$\Delta =9b^2-32ac$

$\Delta =9(a^2+c^2+2ac)-32ac$

$\Delta =9(a^2+c^2+2ac)-32ac$

$\Delta =9a^2+9c^2+18ac-32ac$

$\Delta =9a^2+9c^2-14ac$

$\Delta =7(a^2+c^2-2ab)+2(a^2+c^2)$

$\Delta =7(a-c{)}^2+2(a^2+c^2)$

clearly, $(a-c{)}^2>0$ & $a^2+c^2>0$ $\forall a,c\in R$

thus, $\Delta >0$.

Then equation has real roots.