Question

If $b_1{b}_2=2(c_1+c_2)$, 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. Imaginary roots.
• B. Real roots.
• C. Purely Imaginary roots.
• D. None of the above.

Answer 1

Answer: (B)

Real roots.

$b_1{b}_2=2(c_1+c_2)x^2+b_{1}x+c_1=0D_1={b_1}^2-4c_1{x}^2+b_{2}x+c_2=0D_2={b_2}^2-4c_2{D}_1+D_2={b_1}^2-4c_1+{b_2}^2-4c_2={b_1}^2+{b_2}^2-4{(c}_1+c_2)={b_1}^2+{b_2}^2-2b_1{b}_2={{(b}_1-b_2)}^2\therefore D_1+D_2\ge 0$

for every $x\in R$

so real roots.