Common Roots of Quadratic Equations

Understanding common roots of quadratic equations is a crucial skill for tackling advanced problems in competitive exams like JEE. This topic often appears in sections related to algebra and calculus, requiring a solid grasp of the relationships between roots and coefficients.

What are Common Roots?

When two quadratic equations share at least one root, that root is called a common root. If both roots are common, the equations are essentially multiples of each other. The challenge lies in identifying and utilizing this shared root to solve for unknown coefficients or to find the roots themselves.

Methods to Find Common Roots

There are several algebraic methods to find common roots. One common approach involves using the properties of roots (sum and product) and manipulating the equations to eliminate terms and isolate the common root.

If two quadratic equations have a common root, that root satisfies both equations.

Let the two quadratic equations be $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ . If $\alpha$ is a common root, then $a_1\alpha^2 + b_1\alpha + c_1 = 0$ and $a_2\alpha^2 + b_2\alpha + c_2 = 0$ .

We can use these two equations to find the value of $\alpha$ . One way is to eliminate the $\alpha^2$ term. Multiply the first equation by $a_2$ and the second by $a_1$ : $a_2(a_1\alpha^2 + b_1\alpha + c_1) = 0 \implies a_1a_2\alpha^2 + a_2b_1\alpha + a_2c_1 = 0$ $a_1(a_2\alpha^2 + b_2\alpha + c_2) = 0 \implies a_1a_2\alpha^2 + a_1b_2\alpha + a_1c_2 = 0$ Subtracting the second modified equation from the first gives: $(a_2b_1 - a_1b_2)\alpha + (a_2c_1 - a_1c_2) = 0$ If $a_2b_1 - a_1b_2 \neq 0$ , then $\alpha = \frac{a_1c_2 - a_2c_1}{a_2b_1 - a_1b_2}$ . This expression gives the common root. Substituting this value back into either of the original equations can help solve for unknown coefficients.

What is the condition for two quadratic equations to have a common root, expressed using their coefficients?

The condition for $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ to have a common root is $(c_1a_2 - c_2a_1)^2 = (b_1c_2 - b_2c_1)(a_1b_2 - a_2b_1)$ .

Another powerful method involves using the discriminant and properties of roots. If $\alpha$ is the common root, let the other roots be $\beta$ and $\gamma$ . Then the equations are $(x-\alpha)(x-\beta) = 0$ and $(x-\alpha)(x-\gamma) = 0$ . This implies that the ratio of coefficients must be related.

Method	Description	When to Use
Elimination of $\alpha^2$	Algebraically eliminate the squared term to find the common root.	When coefficients are simple and direct elimination is feasible.
Ratio of Coefficients	If $\alpha$ is common, the equations can be written as $(x-\alpha)(x-\beta)=0$ and $(x-\alpha)(x-\gamma)=0$ . This leads to relationships between coefficients.	Useful for deriving conditions for common roots or when the structure of the equations suggests this approach.
Using Discriminant	Relate the common root to the discriminants of the equations.	Less direct for finding the root itself, more for deriving conditions.

A key insight: If two quadratic equations have a common root, then the expression formed by cross-multiplying their coefficients (after rearranging to match the form $ax^2+bx+c=0$ ) will be zero.

Consider the two quadratic equations: $x^2 + px + q = 0$ and $x^2 + rx + s = 0$ . If they have a common root $\alpha$ , then $\alpha^2 + p\alpha + q = 0$ and $\alpha^2 + r\alpha + s = 0$ . Subtracting these gives $(p-r)\alpha + (q-s) = 0$ . If $p \neq r$ , then $\alpha = \frac{s-q}{p-r}$ . Substituting this back into the first equation yields a condition on $p, q, r, s$ . The condition for a common root can be derived by eliminating $\alpha$ from these two equations, leading to the relationship $(s-q)^2 = (p-r)(qr-ps)$ . This formula is a direct consequence of the algebraic manipulation shown.

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Example Problem

Find the value of $k$ if the equations $x^2 - 5x + 6 = 0$ and $2x^2 + kx - 12 = 0$ have a common root.

First, find the roots of the first equation: $x^2 - 5x + 6 = 0$ . Factoring gives $(x-2)(x-3) = 0$ . So, the roots are $x=2$ and $x=3$ . Now, we check which of these roots is common to the second equation. Case 1: If $x=2$ is the common root, substitute $x=2$ into $2x^2 + kx - 12 = 0$ : $2(2)^2 + k(2) - 12 = 0 \implies 2(4) + 2k - 12 = 0 \implies 8 + 2k - 12 = 0 \implies 2k - 4 = 0 \implies k = 2$ . Case 2: If $x=3$ is the common root, substitute $x=3$ into $2x^2 + kx - 12 = 0$ : $2(3)^2 + k(3) - 12 = 0 \implies 2(9) + 3k - 12 = 0 \implies 18 + 3k - 12 = 0 \implies 3k + 6 = 0 \implies k = -2$ . Therefore, the possible values for $k$ are $2$ and $-2$ .

Advanced Applications

In JEE, problems involving common roots can be more complex, often involving parameters, inequalities, or integration. Understanding the fundamental algebraic manipulations is key to solving these advanced problems efficiently.

Learning Resources

Common Roots of Quadratic Equations - Concepts and Problems(documentation)

Provides a clear explanation of the concept, formulas, and solved examples for finding common roots of quadratic equations.

Quadratic Equations - Common Roots(blog)

Explains the conditions for common roots and offers a step-by-step approach to solving problems related to this topic.

Common Roots of Two Quadratic Equations(documentation)

A resource that delves into the conditions for common roots and presents problems with solutions, often found in competitive exam preparation materials.

JEE Mathematics: Quadratic Equations - Common Roots(video)

A video tutorial explaining the concept of common roots in quadratic equations with illustrative examples suitable for JEE preparation.

Quadratic Equations - Common Roots(documentation)

A discussion forum where users ask and answer questions about common roots, offering diverse perspectives and problem-solving techniques.

Conditions for Common Roots of Quadratic Equations(documentation)

This resource focuses on deriving the conditions for common roots and provides a formula for the same.

Algebraic Manipulation for Common Roots(tutorial)

While not directly on common roots, this tutorial on factoring quadratics is foundational for solving the initial equations.

Properties of Roots of Quadratic Equations(documentation)

A comprehensive overview of quadratic equations, including the properties of roots, which are essential for understanding common roots.

JEE Advanced Mathematics - Quadratic Equations(blog)

A blog post discussing various aspects of quadratic equations relevant to JEE, potentially including common roots.

Common Roots of Polynomials(wikipedia)

Provides a broader context of common roots for polynomials, which can be extended to quadratic equations and offers theoretical background.