First slide

Theory of equations

Question

If $P (x) = {ax}^{2} + bx + c and Q (x) = - {ax}^{2} + dx + c$ where $ac \neq 0,$ then $P (x) . Q (x) = 0$ has at least

Moderate

A

Four real roots
B

Two real roots
C

Four imaginary roots
D

None of these

Solution

Let all four roots are imaginary. Then roots of both equations $P (x) = 0$ and $Q (x) = 0$ are imaginary.
Thus $b^{2} - 4 ac < 0; d^{2} + 4 ac < 0, so b^{2} + d^{2} < 0,$ which is impossible unless $b = 0, d = 0 .$
So, if $b \neq 0$ or $d \neq 0$ at least two roots must be real.
If $b = 0, d = 0,$ we have the equations.
$P (x) = {ax}^{2} + c = 0$ and $Q (x) = - {ax}^{2} + c = 0$
or $x^{2} = - \frac{c}{a}; x^{2} = \frac{c}{a}$ as one of $\frac{c}{a}$ and $- \frac{c}{a}$ must be positive, so two roots must be real.

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