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 atleast one of the equations $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ has

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

Answer 1

The correct option is A real roots

Using the property $A.M\ge G.M$, we get

$\frac{b_1+b_2}{2}\ge \sqrt{b_1{b}_2}$

Squaring above equation we get

$b_1^2+b_2^2+2b_1{b}_2\ge 4b_1{b}_2$

$b_1^2+b_2^2\ge 2b_1{b}_2$

$b_1^2+b_2^2\ge 4(c_1+c_2)$ ...[Substituting the value of $b_1{b}_2$]

$\therefore (b_1^2-4c_1)+(b_2^2-4c_2)\ge 0$

Both the terms cannot be negative since sum of two negative numbers cannot be greater than zero.

Hence, atleast one of $(b_1^2-4c_1)\ge 0$ and $(b_2^2-4c_2)\ge 0$ must be true

Therefore, atleast one of the equation has real roots