Question

If P(x) = $a{x}^2$ + bx + c and Q(x) = $q{x}^2$ + dx + c, where ac 0 then P(x).Q(x) = 0 has

• A. exactly one real root
• B. atleast two real roots
• C. exactly three real roots
• D. all four are real roots

Answer 1

Answer: (B)

atleast two real roots

Given,

$P\left(x\right)=a{x}^2+bx+c$

hence for real roots

$b^2-4ac\ge 0$

$Q\left(x\right)=-a{x}^2+dx+c$

Hence for real roots,

$d^2+4ac\ge 0$

Now $ac\ne 0$

Now if $ac<0$

then

$b^2-4ac>0$

Hence, P(x).Q(x) has atleast two real roots.

If $ac>0$

then $d^2+4ac>0$

Hence $P\left(x\right).Q\left(x\right)$ has atleast two roots.