JEE Quadratic Equations- Advanced Conceptual Understanding

This page will teach you how to master JEE Quadratic Equations up to JEE Advanced level. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

A root of the equation \(ax^2+bx+c=0\) is a number (real or complex), say \(\alpha\), which satisfies the equation i.e. \(a\alpha^2+b\alpha+c=0\). The roots of the quadratic equation \(ax^2+bx+c=0\) with \(a\neq 0\) are given by \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

To have mastery over Quadratic Equations of JEE Advanced, the main concepts you should be confident of are polynomial equations reducible to quadratic equations, algebraic interpretation of Rolle's theorem, intermediate value theorem and analysis of cubic equation with real coefficients.

Polynomial equations reducible to quadratic equations

• An equation of the form \((x-a)(x-b)(x-c)(x-d)=k\)

• An equation of the form \((x-a)(x-b)(x-c)(x-d)=kx^2\), where \(ab=cd\)

• An equation of the form \((x-a)^4+(x-b)^4=k\)

• An equation of the form \(ax^{2n}+bx^n+c=0 \ , a \neq 0 \) and \(n \in \mathbb N\): Substitute \(x^n=y\)

• Reciprocal equations: \(ax^3+bx^2+bx+a=0\) or \(ax^4+bx^3+cx^2+bx+a=0\)

Algebraic interpretation of Rolle's theorem

• Between any two roots of a polynomial equation \(f(x)=0\), there always exists a roots of its derivative \(f'(x)=0\)

• Relation between roots and derivatives

• Some important deductions from Rolle's theorem

Intermediate Value Theorem

• If \(f(x)\) is a polynomial function such that \(f(a) \neq f(b)\), then \(f(x)\) takes every value between \(f(a)\) and \(f(b)\)

• If \(f(a)\) and \(f(b)\) are of opposite signs, then one root of the equation \(f(x)=0\) must lie between \(a\) and \(b\)

Analysis of cubic equation with real coefficients

• Condition for all three real roots of a cubic equation

• Condition for two real roots and one complex root

• Trigonometrical method of solving cubic equation

\[ \begin{array} { l l } A) \, \text{No such value of } \ k \ \text{exists} & \quad \quad \quad \quad \quad & B) \, \frac12 \\ C) \, -\frac12 & & D) \, 1 \\ \end{array} \]

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\[ \begin{array} { l l } A) \, \text{No such value of } \ k \ \text{exists} & \quad \quad \quad \quad \quad & B) \, \frac12 \\ C) \, -\frac12 & & D) \, 1 \\ \end{array} \]

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\[ \begin{array} { l l } A) \, \text{No such value of } \ k \ \text{exists} & \quad \quad \quad \quad \quad & B) \, \frac12 \\ C) \, -\frac12 & & D) \, 1 \\ \end{array} \]

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\[ \begin{array} { l l } A) \, \text{No such value of } \ k \ \text{exists} & \quad \quad \quad \quad \quad & B) \, \frac12 \\ C) \, -\frac12 & & D) \, 1 \\ \end{array} \]

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\[ \begin{array} { l l } A) \, \text{No such value of } \ k \ \text{exists} & \quad \quad \quad \quad \quad & B) \, \frac12 \\ C) \, -\frac12 & & D) \, 1 \\ \end{array} \]

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Once you are confident of Quadratic Equations, move on to JEE Complex Numbers.