Question

If $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2$ are all real numbers, then at least one of the equations $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ has

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

Answer 1

The correct option is C real roots

$b_1{b}_2=2(c_1+c_2)...\left(1\right)$

$x^2+b_{1}x+c_1=0$
$D_1=b_1^2-4c_1$

$x^2+b_{2}x+c_2=0$
$D_2=b_2^2-4c_2$

$D_1+D_2=b_1^2+b_2^2-4(c_1+c_2)=(b_1-b_2{)}^2\ge 0$

Atleast one of them $D_1,D_2$ is greater than or equal to zero.

Hence one of the quadratic must have real roots.