Problem 20 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The value of ' \(c\) ' for which \(\left|\alpha^{2}-\beta^{2}\right|=7 / 4\), where \(\alpha\) and \(\beta\) are the roots of \(2 x^{2}+7 x+c\) \(=0\), is (a) 4 (b) 0 (c) 6 (d) 2

Step-by-Step Solution

Answer

The value of \( c \) is 6.

Step 1: Understand the Roots Relationship

Given a quadratic equation: \(2x^2 + 7x + c = 0\), the roots \( \alpha \) and \( \beta \) are related to the coefficients of the equation by Vieta's formulas. Specifically, \( \alpha + \beta = -\frac{b}{a} = -\frac{7}{2} \) and \( \alpha\beta = \frac{c}{2} \).

Step 2: Express \( \alpha^2 - \beta^2 \) in another form

The expression \( \alpha^2 - \beta^2 \) can be rewritten using the identity \( \alpha^2 - \beta^2 = (\alpha + \beta)(\alpha - \beta) \).

Step 3: Substitute Vieta's Formula

Substitute \( \alpha + \beta = -\frac{7}{2} \) into \( (\alpha + \beta)(\alpha - \beta) \) to get: \((\alpha + \beta)(\alpha - \beta) = -\frac{7}{2}(\alpha - \beta)\).

Step 4: Solve for \( \alpha - \beta \)

Since \( \left|\alpha^2 - \beta^2\right| = 7/4 \), we have \( \left| -\frac{7}{2}(\alpha - \beta) \right| = \frac{7}{4} \). This simplifies to \( \frac{7}{2} |\alpha - \beta| = \frac{7}{4} \).

Step 5: Simplify and Solve for \( c \)

Divide both sides by \( \frac{7}{2} \) to find \(|\alpha - \beta| = \frac{1}{2}\). The discriminant of the quadratic is given by \( b^2 - 4ac \), where \( \alpha - \beta = sqrt{\frac{b^2 - 4ac}{a}} \). Set \( \sqrt{7^2 - 8c} = 1 \), solve for \( c \).

Step 6: Calculate Discriminant

Solve \( \sqrt{49 - 8c} = 1 \), square both sides to obtain \( 49 - 8c = 1 \), and solve for \( c \).

Step 7: Final Calculation

Subtract 1 from 49 to get \(48 = 8c\), divide by 8 to find \(c = 6\).

Key Concepts

Vieta's formulas

In the world of quadratic equations, Vieta's formulas provide a fascinating way to relate the roots of the equation to its coefficients. Simply put, if a quadratic equation is of the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us that:

• \( \alpha + \beta = -\frac{b}{a} \)
• \( \alpha \beta = \frac{c}{a} \)

Here, \( \alpha \) and \( \beta \) are the roots of the quadratic equation.

These formulas come in handy when you wish to find the sum and the product of the roots without actually solving the equation. For example, in the given exercise with the quadratic \( 2x^2 + 7x + c = 0 \), Vieta's formulas give us that \( \alpha + \beta = -\frac{7}{2} \) and \( \alpha \beta = \frac{c}{2} \). This linkage shows how the coefficients directly reveal the characteristics of the roots, making problem-solving more efficient and elegant.

roots of quadratic equation

The roots of a quadratic equation are the values of \( x \) that make the equation true. In basic terms, if you set your quadratic equation to zero, the solutions \( x \) are your roots. Knowing that, it becomes clear that the relationship between the roots \( \alpha \) and \( \beta \) is crucial. You can think of these roots in terms of their expressions using Vieta's formulas, but there's more.

Every quadratic equation has a specific formula to find these roots called the "quadratic formula," which is given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]This formula directly relates the roots to the coefficients of the equation. To uncover the roots, you will need to substitute the values for \( a \), \( b \), and \( c \) from your specific equation.

In our case with \( 2x^2 + 7x + c = 0 \), finding the values of \( \alpha \) and \( \beta \) can help further analyze the relationships indicated by Vieta's formulas. By finding these roots, you can better navigate the relationships between them, including expressions like \( \alpha^2 - \beta^2 \). This step bridges directly into understanding how these roots satisfy other conditions provided, such as in the given problem.

discriminant calculation

One of the essential parts of solving a quadratic equation with the quadratic formula is understanding the discriminant. The discriminant, denoted as \( b^2 - 4ac \), is what sits inside the square root part of the formula. It determines both the nature and the number of roots of your quadratic equation.

Here's how it works:

• If the discriminant \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), no real roots exist, only complex or imaginary ones.

For the problem at hand, we manipulate the discriminant \( b^2 - 4ac \) under the constraint \( |\alpha - \beta| = \frac{1}{2} \), derived from \( |\alpha^2 - \beta^2| = \frac{7}{4} \).

Solving \( \sqrt{49 - 8c} = 1 \), for example, involves rearranging and calculating the discriminant to ensure that condition is met. This step means squaring both sides to get \( 49 - 8c = 1 \) and finding \( c = 6 \). Understanding discriminant calculation is key to discerning the behavior of the roots in any quadratic equation.