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4-2.Quadratic Equations and Inequations
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Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?
A
If $a^2-2 b < 0$, then the equation has one real and two imaginary roots
B
If $a^2-2 b \geq 0$, then the equation has all real roots.
C
If $a^2-2 b > 0$, then the equation has all real and distinct roots.
D
If $4 a^3-27 b^2 > 0$, then the equation has real and distinct roots.
(KVPY-2011)
Solution
(a)
We have, $x^3+a x^2+b x+c=0$
Let $f(x)=x^3+a x^2+b x+c$
$P^{\prime}(x) =3 x^2+2 a x+b$
$D =(2 a)^2-4(3)(b)$
$D =4 a^2-12 b$
$D =4\left(a^2-3 b\right)$
For three roots $a^2-3 b > 0$
but given $a^2-2 b < 0$
$\therefore \quad D < 0$
Hence, $f(x)$ has one real roots and two imaginary roots.
Standard 11
Mathematics
