Question

If $b_1b_2 = 2(c_1 + c_2)$ and $b_1, b_2, c_1, c_2$ are all real numbers, then at least one of the equations $x^2 + b_1x + c_1 = 0$ and $x^2 + b_2x + c_2 = 0$ has

• A. real roots
• B. purely imaginary roots
• C. roots of the form $a + ib (a, b \in \, R , ab \neq 0)$
• D. rational roots

Answer 1

Answer: (A)

We have equations
and Now, $x^{2}+b_{1} x+c_{1}=0$
$D_{1}=b_{1}^{2}-4 c_{1}$
and $x^{2}+b_{2} x+c_{2}=0$
$D_{2}=b_{2}^{2}-4 c_{2}$
Now, $D_1 + D_{2}=b_{1}^{2}+b_{2}^{2}-4\left(c_{1}+c_{2}\right)$
$=b_{1}^{2}+b_{2}^{2}-2 b_{1} b_{2} \left[\because b_{1} b_{2}=2\left(c_{1}+c_{2}\right)\right]$
$=\left(b_{1}-b_{2}\right)^{2} \geq 0$
$\Rightarrow $ At least one of $D_{1}$ and $D_{2}$ are non-negative real roots.