Problem 5 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If one root of the equation \(5 x^{2}+13 x+k=0\) is reciprocal of other, then find the value of \(k\).

Step-by-Step Solution

Answer

The value of \( k \) is 5.

Step 1: Identifying the Relationship

Since one root is the reciprocal of the other, let's denote the roots of the quadratic equation by \(r\) and \(\frac{1}{r}\). From the theory of quadratic equations, we know that the product of the roots \(r \times \frac{1}{r} = 1\).

Step 2: Using the Product of the Roots Formula

According to Vieta's formulas for a quadratic equation \(ax^{2} + bx + c = 0\), the product of the roots \(\frac{c}{a}\) equals the constant term \(c\) divided by the leading coefficient \(a\). Here, \(c = k\) and \(a = 5\), so we have: \[ r \times \frac{1}{r} = \frac{k}{5} = 1 \]

Step 3: Solving for k

Since \(\frac{k}{5} = 1\), we can solve for \(k\) by multiplying both sides by 5: \[ k = 5 \]

Step 4: Verifying the Solution with Relationship

Now, substitute \(k = 5\) in the equation to verify: The equation is \(5x^2 + 13x + 5 = 0\). The roots should satisfy the requirement of being reciprocals. If \(k = 5\), the product of roots is \(\frac{5}{5} = 1\), confirming the relationship \(r \times \frac{1}{r} = 1\).

Key Concepts

Roots of Quadratic Equation

Quadratic equations are polynomials of degree 2, typically represented in the form \( ax^2 + bx + c = 0 \). The "roots" refer to the solutions for \( x \) that satisfy the equation. These roots can be real or complex numbers. For example, in the equation \( 5x^2 + 13x + k = 0 \), the task is to find these values for which the equation holds true.

The roots of a quadratic equation can often be found using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The part under the square root sign, known as the discriminant, determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if zero, there is exactly one real root (a repeated root); and if negative, the roots are complex (non-real).

Understanding roots is critical because they offer insight into the behavior of the equation's graph, such as where it intersects the x-axis. Knowing that one root is the reciprocal of the other, as in this problem, simplifies determining the constant \( k \).

Reciprocal Roots

Reciprocal roots mean that if one root is \( r \), the other is \( \frac{1}{r} \). This relationship simplifies many equations because the product of complementary reciprocals is always 1:

• \( r \times \frac{1}{r} = 1 \)

This problem utilizes reciprocal roots by introducing a special condition where the roots of the quadratic equation are reciprocals. This simplifies finding the value of \( k \) in our equation since the product of the roots can be directly related to this concept.

Given a quadratic equation \( ax^2 + bx + c = 0 \), if we know that the product of the roots \( r \times \frac{1}{r} = 1 \), we can verify this condition using Vieta's formulas, which leads us to find the unknown constant \( c \) in terms of \( k \).

Vieta's Formulas

Vieta's Formulas provide a quick way to relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \( ax^2 + bx + c = 0 \), these formulas say:

• Sum of the roots (\( r_1 + r_2 = -\frac{b}{a} \))
• Product of the roots (\( r_1 \cdot r_2 = \frac{c}{a} \))

In our particular problem, the relationship focuses on the product of the roots. Having reciprocal roots implies \( r \cdot \frac{1}{r} = 1 \). Plugging this into the product formula \( r_1 \cdot r_2 = \frac{k}{5} \) results in \( \frac{k}{5} = 1 \). Solving for \( k \) gives the important conclusion \( k = 5 \).

Vieta's Formulas thus provide an elegant and efficient way to derive foundational relationships between the terms of a polynomial and the nature of its solutions. They are not only helpful in finding certain coefficients like \( k \), but also in verifying the correctness of solutions.