Problem 34 - A Complete Resource Book in Mathematics for JEE Main

Question

If the roots of the equation \(a x^{2}+b x+c=0\) are of the form \(\frac{\alpha}{\alpha-1}\) and \(\frac{\alpha+1}{\alpha}\), then the value of \((a+b+c)^{2}\) is (A) \(b^{2}-2 a c\) (B) \(b^{2}-4 a c\) (C) \(4 b^{2}-a c\) (D) \(2 b^{2}-a c\)

Step-by-Step Solution

Answer

The value of \((a + b + c)^2\) is \(b^2 - 4ac\).

Step 1: Identify the Roots

We are given that the roots of the quadratic equation are \(\frac{\alpha}{\alpha-1}\) and \(\frac{\alpha+1}{\alpha}\). We need to use these roots in relation to the quadratic equation \(ax^2 + bx + c = 0\).

Step 2: Apply Vieta's Formulas

According to Vieta's formulas, the sum of the roots \(\left( \frac{\alpha}{\alpha-1} + \frac{\alpha+1}{\alpha} \right)\) must equal \(\frac{-b}{a}\) and the product of the roots \(\left( \frac{\alpha}{\alpha-1} \cdot \frac{\alpha+1}{\alpha} \right)\) must equal \(\frac{c}{a}\).

Step 3: Calculate the Sum of the Roots

Calculate the sum of the given roots: \[ \frac{\alpha}{\alpha-1} + \frac{\alpha+1}{\alpha} = \frac{\alpha^2 + (\alpha-1)(\alpha+1)}{\alpha(\alpha-1)} = \frac{2\alpha^2-1}{\alpha^2-\alpha}. \] Let this equal \(\frac{-b}{a}\).

Step 4: Calculate the Product of the Roots

Now, calculate the product of the roots: \[ \frac{\alpha}{\alpha-1} \cdot \frac{\alpha+1}{\alpha} = \frac{\alpha^2+\alpha}{\alpha(\alpha-1)} = \frac{\alpha+1}{\alpha-1}. \] Let this equal \(\frac{c}{a}\).

Step 5: Find Relationships Between Coefficients

Given that \(a\times\) sum = \(-b\) and \(a\times\) product = \(c\), solve these equations to determine relationships among \(a, b,\) and \(c\). Simplify the relationships using common mathematical operations.

Step 6: Calculate \((a+b+c)^2\)

Use the relationships derived in the previous step to calculate \((a + b + c)^2\). Find any simplifications by substituting \(a, b, c\) values as calculated.

Key Concepts

Vieta's formulas

Vieta's formulas provide a powerful tool to relate the coefficients of a polynomial equation to its roots. For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas tell us:

• The sum of the roots \(x_1 + x_2 = -\frac{b}{a}\).
• The product of the roots \(x_1 \cdot x_2 = \frac{c}{a}\).

By using these relationships, one can find connections between the roots and their corresponding coefficients, which can simplify problem-solving without actually finding the roots themselves.
Vieta's formulas are immensely helpful in exercises involving specific roots forms, like the given problem with unconventional fractions. They transform the task into simpler algebraic manipulations using the equation's coefficients.

Sum of Roots

The sum of the roots of a quadratic equation can be easily determined using Vieta’s formulas. In the equation \(ax^2 + bx + c = 0\), the sum of the roots \(x_1 + x_2\) is given by \[x_1 + x_2 = -\frac{b}{a}\]This is derived directly from rearranging the quadratic formula and shows the linear relationship between the sum of the roots and the coefficients.
In the original problem, after finding the expressions for the roots, you calculate the sum \[\frac{\alpha}{\alpha-1} + \frac{\alpha+1}{\alpha} = \frac{2\alpha^2-1}{\alpha^2-\alpha}\]and equate it to \(-\frac{b}{a}\). This step is crucial for unveiling relationships among the coefficients \(a\), \(b\), and \(c\).

Product of Roots

Like the sum of roots, Vieta's formulas also provide a straightforward way to find the product of roots for the quadratic equation \(ax^2 + bx + c = 0\). The formula states:\[x_1 \cdot x_2 = \frac{c}{a}\]This product is a direct expression of the relationship between the constant term \(c\) and the leading coefficient \(a\).
In our problem, the product of the roots is calculated as:\[\frac{\alpha}{\alpha-1} \cdot \frac{\alpha+1}{\alpha} = \frac{\alpha+1}{\alpha-1}\]This product is then set equal to \(\frac{c}{a}\), helping identify how the coefficients interact and solve for specific terms.

Relationship Between Coefficients

The relationship between the coefficients of a quadratic equation and its roots is an essential aspect of understanding quadratic equations. In the equation \(ax^2 + bx + c = 0\), each coefficient plays a unique role:

• \(a\) affects the width and direction of the parabola.
• \(b\) influences the vertex and axis of symmetry.
• \(c\) is the y-intercept.

In problems like the one provided, where the roots have specific forms, understanding how the coefficients relate through Vieta's formulas leads to an equation system detailing how \(a, b, c\) interact.
Step 5 of the solution involved using the derived sum and product of roots to create equations that directly reflect these relationships, allowing further deduction of \((a + b + c)^2\) and simplification into one of the given choices.