(A) Let $f(x) = x^3+ax^2+bx+c$. Then $f'(x) = 3x^2+2ax+b$. The discriminant of the quadratic $f'(x)$ is $D = (2a)^2 - 4(3)(b) = 4a^2 - 12b = 4(a^2-3b)

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If $a^2-2b < 0$,then the equation has one real and two imaginary roots.

(A) Let $f(x) = x^3+ax^2+bx+c$. Then $f'(x) = 3x^2+2ax+b$. The discriminant of the quadratic $f'(x)$ is $D = (2a)^2 - 4(3)(b) = 4a^2 - 12b = 4(a^2-3b)$. If $a^2-2b Since $b$ must be positive for $a^2 Thus,$D 0$ for all $x$. Since $f'(x)$ is always positive,$f(x)$ is a strictly increasing function. $A$ strictly increasing cubic polynomial has exactly one real root and two complex conjugate roots. Therefore,the statement in option $A$ is correct.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Solution of quadratic equations and Nature of roots

Consider the cubic equation $x^3+ax^2+bx+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?

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A
If $a^2-2b < 0$,then the equation has one real and two imaginary roots.
B
If $a^2-2b \geq 0$,then the equation has all real roots.
C
If $a^2-2b > 0$,then the equation has all real and distinct roots.
D
If $4a^3-27b^2 > 0$,then the equation has real and distinct roots.

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4-2.Quadratic Equations and Inequations

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Solution of quadratic equations and Nature of roots

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Consider the cubic equation $x^3+ax^2+bx+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?

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