Problem 138 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then their other roots will satisfy the equation \(x^{2}+a x+b c=0\) Reason: If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+\) \(c x+a b=0\) have a common root, then \(a+b+c=0\)

Step-by-Step Solution

Answer

The assertion is true; the roots satisfy the equation if \(a + b + c = 0\).

Step 1: Identify the Equations

We are given two quadratic equations: \( x^2 + bx + ca = 0 \) and \( x^2 + cx + ab = 0 \). We need to find a common root of these equations and determine the truth of the assertion.

Step 2: Assume a Common Root

Let \( \alpha \) be the common root of both equations. Thus, substituting \( \alpha \) in both equations, we have: \( \alpha^2 + b\alpha + ca = 0 \) and \( \alpha^2 + c\alpha + ab = 0 \).

Step 3: Subtract the Equations

By subtracting the second equation from the first, we obtain: \((b-c)\alpha + ca - ab = 0\).

Step 4: Analyze the Common Root Condition

For the above equation to hold, given \(\alpha\) is a common root, if \(b eq c\), then \(\alpha = \frac{ab-ca}{b-c}\). Now, solving both original equations for the other root in terms of \(a\), \(b\), and \(c\), we use the condition obtained from the subtraction step.

Step 5: Relation to the Assertion

Utilizing the condition for a common root \( a + b + c = 0 \), substitute this into the equation \( x^2 + ax + bc = 0 \) to see if the other roots of the original equations satisfy it.

Step 6: Verify the Relation

Substituting \( a + b + c = 0 \) into the new quadratic equation results in \( x^2 + (-b-c)x + bc = 0 \). Solving the original equations under the condition \( a + b + c = 0 \) ensures the roots are consistent.

Key Concepts

Common Roots

In algebra, when two quadratic equations are said to have a common root, it means there is at least one solution that satisfies both equations. This commonality provides a connection between the equations and can sometimes hint towards deeper relationships or simplifications.
To find a common root of the equations \(x^2 + bx + ca = 0\) and \(x^2 + cx + ab = 0\), we typically start by assuming such a root exists. Let's call it \( \alpha \). This means \( \alpha \) satisfies both equations:

• \( \alpha^2 + b\alpha + ca = 0 \)
• \( \alpha^2 + c\alpha + ab = 0 \)

Subtracting the second equation from the first, we obtain an expression that contains only \( \alpha \), \(b\), \(c\), \(a\), and constants: \( (b-c)\alpha + ca - ab = 0 \). This allows for further exploration of the conditions under which \( \alpha \) can be a common root.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
This formula allows us to calculate the roots directly, as long as we correctly identify the coefficients \(a\), \(b\), and \(c\).
By applying this formula to the quadratic equations given in this exercise (\(x^2 + bx + ca = 0\) and \(x^2 + cx + ab = 0\)), one can determine not only the common root but also the other roots of the equations.
In scenarios where the equations have a common root, substituting \(b\), \(c\), and \(a\) into the quadratic formula can affirm the assertion that a particular relationship, such as \(a + b + c = 0\), holds true. Understanding how to use the quadratic formula in this context further elucidates the relationships between the involved terms.

Algebraic Manipulation

Algebraic manipulation plays a crucial role in simplifying and solving equations. In the given exercise, once we identify that the equations \(x^2 + bx + ca = 0\) and \(x^2 + cx + ab = 0\) have a common root \( \alpha \), our task is to elegantly manipulate these equations.

• Start by setting equal the derived expressions from substituting \( \alpha \).
• Next, subtract one equation from the other to eliminate the \( \alpha^2 \) term, leading to \( (b-c)\alpha + ca - ab = 0 \).

If \(b\) is not equal to \(c\), solve for \( \alpha \) to find the potential common root. When given conditions such as \(a+b+c=0\), substitution into newly formed equations (like \(x^2 + ax + bc = 0\)) helps verify if they form a true statement.
Algebraic manipulations allow transition from complex, multi-variable expressions into simpler forms, often revealing integral truths about the relationships between equations.