First slide

Theory of equations

Question

$If f (x) = a x^{2} + b x + c, g (x) = - a x^{2} + b x + c where a c \neq 0 then f (x) g (x) = 0 has$

Moderate

A

at least two real roots
B

two real roots and two imaginary roots
C

at least three real roots
D

no real roots

Solution

$I et D_{1} {and D}_{2} be discriminants of a x^{2} + b x + c - 0 and - a x^{2} + b x + c = 0 respectively$
$Then, D_{1} = b^{2} - 4 a c, D_{2} = b^{2} + 4 a c$
$Now, a c \neq 0 \Rightarrow either a c > 0 or a c < 0$
$If a c > 0, then D_{2} > 0 . Therefore, roots of - a x^{2} + b x + c = 0 a r e r e a l$
$If a c < 0, then D_{1} > 0 . Therefore, roots of a x^{2} + b x + c = 0 are real$
$Thus, f (x) g (x) has at least two real roots.$

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