Problem 7 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If 8 and 2 are roots of the quadratic equation \(x^{2}+\alpha x+\beta=0\) and 3,3 are roots of the quadratic equation \(x^{2}+\alpha x+b=0 .\) Then, find the roots of the quadratic equation \(x^{2}+a x+b=0\)

Step-by-Step Solution

Answer

The roots of the equation \(x^2 + ax + b = 0\) are 3 and 3.

Step 1: Apply Vieta's Formulas to First Quadratic Equation

For the equation \(x^2 + \alpha x + \beta = 0\) with roots 8 and 2, Vieta’s formulas tell us: \(\alpha = -(8 + 2) = -10\) and \(\beta = 8 \times 2 = 16\).

Step 2: Apply Vieta's Formulas to Second Quadratic Equation

For the equation \(x^2 + \alpha x + b = 0\) with roots 3 and 3, we have: \(\alpha = -(3 + 3) = -6\) and \(b = 3 \times 3 = 9\).

Step 3: Use Identified Values for the Third Equation

We now substitute \(a = \alpha = -6\) and \(b = 9\) into the third quadratic equation: \(x^2 + ax + b = 0\). So, the equation becomes \(x^2 - 6x + 9 = 0\).

Step 4: Solve the Simplified Quadratic Equation

Solve \(x^2 - 6x + 9 = 0\) using factorization: \((x - 3)^2 = 0\). Therefore, the root is \(x = 3\) with a multiplicity of 2.

Key Concepts

Vieta's Formulas and Their Application

Vieta’s formulas provide a bridge between the coefficients of a polynomial and the roots of that polynomial. Specifically, they relate sums and products of the roots to the coefficients.

• The sum of the roots of the quadratic equation is given by \(-\frac{b}{a}\), where \(a\) and \(b\) are coefficients in the equation \(ax^2 + bx + c = 0\).
• The product of the roots is \(\frac{c}{a}\).

For example, when applying Vieta's formulas to the equation \(x^2 + \alpha x + \beta = 0\) with roots 8 and 2:

• The sum is: \(\alpha = -(8 + 2) = -10\).
• The product is: \(\beta = 8 \times 2 = 16\).

Vieta's formulas simplify the process of finding the relationships between the roots and coefficients, making it easier to solve quadratic equations.

Understanding Quadratic Roots

Quadratic roots are the values of \(x\) for which the quadratic equation \(ax^2 + bx + c = 0\) equals zero. These roots can be real or complex, but in most real-world applications, we deal with real roots.Finding these roots often involves different techniques:

• Factoring: This method involves writing the quadratic as a product of two binomials.
• Using the Quadratic Formula: This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), works for all types of quadratic equations.
• Completing the square: This involves transforming the quadratic into a perfect square trinomial.

For example, solving the equation \(x^2 - 6x + 9 = 0\) by factoring gives \((x - 3)^2 = 0\), indicating that \(x = 3\) is a root with a multiplicity of 2, meaning the solution is repeated.

Factorization of Quadratic Equations

Factorization is an efficient method to solve quadratic equations, especially when the roots are simple integers. In factorization, you express the quadratic expression as a product of two binomials.
For the quadratic equation \(x^2 - 6x + 9 = 0\), factorization gives:

• Identify two numbers that multiply to 9 (the constant term) and add up to -6 (the coefficient of \(x\)).
• These numbers are 3 and 3.

Thus, the quadratic can be rewritten as: \((x - 3)(x - 3) = (x - 3)^2 = 0\).
Solving this equation results in \(x = 3\), which is the only root of the equation but appears twice. This illustrates the concept of a repeated root or root of multiplicity.