Problem 18 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\alpha, \beta\) are the roots of the quadratic equation \(x^{2}+b x-c=0\), then the equation whose roots are \(b\) and \(c\) is (a) \(x^{2}+\alpha x-\beta=0\) (b) \(x^{2}-[(\alpha+\beta)+\alpha \beta] x-\alpha \beta(\alpha+\beta)=0\) (c) \(x^{2}-[(\alpha+\beta)+\alpha \beta] x+\alpha \beta(\alpha+\beta)=0\) (d) \(x^{2}+[\alpha \beta+(\alpha+\beta)] x-\alpha \beta(\alpha+\beta)=0\)

Step-by-Step Solution

Answer

(c) \(x^2 - [(\alpha+\beta) + \alpha \beta]x + \alpha \beta (\alpha+\beta) = 0\) is correct.

Step 1: Identify the given equation and its roots

The given quadratic equation is \(x^{2}+bx-c=0\), and its roots are \(\alpha\) and \(\beta\). Based on Vieta's formulas, we have \(\alpha + \beta = -b\) and \(\alpha \beta = -c\).

Step 2: Construct the new equation

We need to find an equation whose roots are \(b\) and \(-c\). Let the new equation be \(x^2 - (\text{sum of the roots})x + (\text{product of the roots}) =0\).

Step 3: Calculate the sum of new roots

The sum of the new roots \(b\) and \(-c\) is \(b + (-c) = b - c\).

Step 4: Calculate the product of new roots

The product of the new roots \(b\) and \(-c\) is \(b \cdot (-c) = -bc\).

Step 5: Formulate the new equation

Based on the sum and product of the roots, the new equation is \(x^2 - (b-c)x - bc = 0\).

Step 6: Match with given options

After comparing our formulated equation \(x^2 - (b-c)x - bc = 0\) with the given options, we see that none of the options match. Hence, let's express \(b\) and \(c\) using \(\alpha\) and \(\beta\): \(b = - (\alpha + \beta)\) and \(c = \alpha \beta \). Substituting these values gives us \(x^2 - (-(\alpha + \beta) - \alpha \beta)x + \alpha \beta (-(\alpha + \beta) ) =0\). This simplifies to \(x^2 - [(\alpha+\beta) + \alpha \beta]x + \alpha \beta (\alpha+\beta) = 0\).

Step 7: Determine correct answer

The simplified equation matches option (c): \(x^2 - [(\alpha+\beta) + \alpha \beta]x + \alpha \beta (\alpha+\beta) = 0\).

Key Concepts

Vieta's Formulas

Vieta's Formulas are a key concept in understanding the relationship between the coefficients and roots of a polynomial equation. For a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's Formulas state:

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

In the exercise, we have a quadratic equation \( x^2 + bx - c = 0 \) whose roots are \( \alpha \) and \( \beta \). Hence, according to Vieta:

• \( \alpha + \beta = -b \)
• \( \alpha \beta = -c \)

These relationships allow us to express the coefficients of an equation using its roots, which is essential for transforming equations and solving problems that involve changing the roots.

Roots of Equations

In a quadratic equation, the term 'roots' refers to the solutions for the variable where the equation equals zero. Suppose we have a quadratic equation in the general form \( x^2 + px + q = 0 \). The roots \( \alpha \) and \( \beta \) are the values that satisfy the equation when substituted back into it. These values are slightly different in every equation depending on the specific coefficients.

When a problem asks for a new quadratic equation with different roots, like \( b \) and \(-c \), we use the new roots to form a new equation. The new equation still follows Vieta's designs where:

• Sum of the roots \( b \) and \(-c \) gives us \( b + (-c) = b - c \).
• Product of the roots \( b \) and \(-c \) gives us \(-bc \).

These calculations serve as the foundation for creating a new equation whose roots are rearranged based on manipulations of the original quadratic equation.

Equation Transformation

Equation transformation involves changing an equation's structure by altering its roots or coefficients, which is crucial for some algebraic manipulations needed in exercises like the one above.

In the given task, we're required to find an equation with roots \( b \) and \(-c \) when starting with the roots \( \alpha \) and \( \beta \). This involves a series of simple steps:

• First, identify the sum and product of the new roots as \( b-c \) and \(-bc \) respectively.
• Use these to form a new quadratic equation in the standard form \( x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \).
• Finally, if these new roots relate back to \( \alpha \) and \( \beta \), we can express \( b \) and \( c \) with respect to \( \alpha + \beta \) and \( \alpha \beta \), yielding transformations that align the new equation with any constraints noted in the question.

This transformational process is an essential skill for any algebra student as it teaches logical manipulation of algebraic expressions to reach desired solutions.