Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are rational and \(a x^{2}+b x+c=0\) and \(3 x^{2}+\) \(x-5=0\) have a common root, then \(3 a+b+2 c=\) (A) 0 (B) 1 (C) 2 (D) None of these

Step-by-Step Solution

Answer

Option (A) 0

Step 1: Compare Coefficients

Given that the equation \(3x^2 + x - 5 = 0\) and \(ax^2 + bx + c = 0\) have a common root. Let's assume the common root is \(\alpha\). According to the problems, the equation \(ax^2 + bx + c = 0\) is also satisfied by \(\alpha\).

Step 2: Apply Vieta's Formulas

Using Vieta's formulas for the first equation \(3x^2 + x - 5 = 0\), we know that the sum of roots \((\alpha_1 + \alpha_2)\) is \(-\frac{1}{3}\) and product of roots \((\alpha_1 \cdot \alpha_2)\) is \(-\frac{5}{3}\). Therefore, if \(\alpha\) is a root, then the other root is \(\frac{-1}{3} - \alpha\).

Step 3: Substitute Common Root Into Second Equation

Substitute \(\alpha\) into the equation \(ax^2 + bx + c = 0\), which gives:\(a\alpha^2 + b\alpha + c = 0\). Since \(\alpha\) is also a root of the equation \(3x^2 + x - 5 = 0\), then \(3\alpha^2 + \alpha - 5 = 0\).

Step 4: Derive System of Equations

For \(ax^2 + bx + c = 0\) to have a common root with \(3x^2 + x - 5 = 0\), there exists a relationship between the coefficients. By expressing \(a, b, c\) in terms of 3, 1, -5, we can derive a system that satisfies this condition. For simplicity:\(a = 3k\),\(b = k\),\(c = -5k\), for some constant \(k\).

Step 5: Solve for Given Expression

Plug \(a = 3k\), \(b = k\), and \(c = -5k\) into the expression \(3a + b + 2c\):\(3(3k) + k + 2(-5k) = 9k + k - 10k = 0\).

Step 6: Conclude the Result

The expression \(3a + b + 2c = 0\) confirms option (A) as the correct choice.

Key Concepts

Quadratic Equations

Quadratic equations are one of the fundamental concepts in algebra. They are polynomial equations of degree two, typically expressed in the standard form: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations can have either two distinct real roots, one real root, or two complex roots.

Quadratic equations can be solved using several methods:

• Factoring: If the quadratic can be factored easily, this method simplifies the solution process by setting each factor to zero.
• Quadratic Formula: The quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]provides a clear method to find the roots, regardless of their nature (real or complex).
• Completing the Square: This transforms the quadratic equation into a perfect square trinomial, making it easier to solve.

Understanding these methods is key for solving many equations in mathematics and offers a foundation for studying more complex algebraic concepts.

Vieta's Formulas

Vieta's formulas offer a remarkable relationship between the coefficients of a polynomial and its roots. These formulas are particularly useful for quadratic equations, simplifying many aspects of algebraic solutions. For a quadratic equation \[ ax^2 + bx + c = 0 \],Vieta's formulas state:

• The sum of the roots \( \alpha + \beta \) is \( -\frac{b}{a} \).
• The product of the roots \( \alpha \cdot \beta \) is \( \frac{c}{a} \).

These relationships are derived from the structure of the quadratic equation and are immensely helpful in solving problems without explicitly finding the roots.

In our exercise, Vieta's formulas were used to express the roots of the equation \(3x^2 + x - 5 = 0\), assisting in deducing the common root with another equation. Using Vieta's insights reduces the computational complexity and enhances problem-solving proficiency.

Rational Roots

Rational roots are solutions to polynomial equations that can be expressed as a ratio of two integers, \( \frac{p}{q} \), where \(q eq 0\). In both quadratic and other polynomial equations, determining whether roots are rational can simplify the problem considerably.

The \[\text{Rational Root Theorem}\] provides a criterion for testing potential rational roots of a polynomial equation. Specifically for a polynomial, it posits:

• Any rational root, \( \frac{p}{q} \), is such that \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.

Applying this theorem allows us to generate a finite list of potential rational roots to check. In simpler equations, this can lead directly to a solution or aid in narrowing down possibilities.

Common Roots in Equations

Common roots in equations occur when two or more polynomial equations share at least one solution or root. Identifying common roots is essential in solving systems of polynomial equations, as it denotes a point of intersection between their graphs.

To find common roots:

• First, assume a root \( \alpha \) is common to both equations.
• Substitute \( \alpha \) in each equation and then equate the results to form a system of equations.
• Solve these equations simultaneously to find \( \alpha \) along with any necessary relations between coefficients.

In our exercise, we assumed the existence of a common root \( \alpha \) for the quadratic equations \(ax^2 + bx + c = 0\) and \(3x^2 + x - 5 = 0 \). By equating the values derived from substituting this \( \alpha \), we discovered a relation between the coefficients, enabling us to simplify and solve the problem, showcasing the power of this concept in mathematical analysis.