Question

If $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ are the two quadratic equations such that $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2\in R,$ then

• A. Both the equations will have imaginary roots
• B. at least one equation has real roots
• C. Both the equations will have real roots
• D. at least one equation has imaginary roots

Answer 1

The correct option is B at least one equation has real roots

Let $D_1$ and $D_2$ be discriminants of $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0,$ respectively. Then, $D_1+D_2=b_1^2-4c_1+b_2^2-4c_2$
$=(b_1^2+b_2^2)-4(c_1+c_2)$
$=b_1^2+b_2^2-2b_1{b}_2$
$=(b_1-b_2{)}^2\ge 0$
$\Rightarrow$ At least one of $D_1,D_2$ is non-negative.
Hence, at least one of the equations has real roots.