Question

If $P\left(x\right)={ax}^2+bx+c$ and $Q\left(x\right)={-ax}^2+dx+c$, where $ac\ne 0$, then $P\left(x\right),Q\left(x\right)=0$

• A. Exactly one real root
• B. At least two real roots
• C. Exactly three real roots
• D. All four are real roots

Answer 1

The correct option is B At least two real roots

$P\left(X\right)=a{x}^2+bx+c\to D_1=b^2-4ac$

$Q\left(x\right)=-a{x}^2+dx+c\to D_2=d^2+4ac$

$\{\begin{array}{c}if4ac<0A=b^2-4acisalwaysgreaterthan0\\ If4ac>0D_2=d^2+4acisalwaysgreaterthan0\end{array}$

So, $P\left(x\right),Q\left(x\right)=0$ have at least two real roots.