Question

Suppose $a,b,c$ are real numbers, and each of the equations $x^2+2ax+b^2=0$ and $x^2+2bx+c^2=0$ has two distinct real roots. Then the equation $x^2+2cx+a^2=0$ has

• A. two distinct positive real roots
• B. two equal roots
• C. one positive and one negativeroot
• D. no real roots

Answer 1

The correct option is D no real roots

As the first two equations has real roots ;

$\Rightarrow 4a^2-b^4>0\Rightarrow a^2>b^4/4$

$\Rightarrow 4b^2-c^4>0\Rightarrow b^2>c^4/4\Rightarrow b^2/4>c^4/16$

Combining both equations, we get $a^2>c^4/16$, also $a^4>c^8/256$

Discriminant of 3rd equation is :$\Rightarrow 4c^2-a^4$

As $a^4>c^8/256$ which is much larger than $4c^2$,descriminant is negative

$\Rightarrow$no real roots for 3rd equation