Question
one root of the equation \( 2 x^{2}+5 x+k=0 \) is reciprocal to each other, then \( k \) is
Ask by Moran Dawson. in India
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( k = 2 \)
Solution
Given the quadratic equation:
\[
2x^2 + 5x + k = 0
\]
Let's assume the roots of this equation are \( r \) and \( \frac{1}{r} \) since one root is the reciprocal of the other.
**Using Vieta's Formulas:**
1. **Sum of the roots:**
\[
r + \frac{1}{r} = -\frac{b}{a} = -\frac{5}{2}
\]
2. **Product of the roots:**
\[
r \cdot \frac{1}{r} = \frac{c}{a} = \frac{k}{2}
\]
\[
1 = \frac{k}{2} \implies k = 2
\]
**Verification:**
To ensure that the roots are real, the discriminant must be non-negative:
\[
\Delta = b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9 \geq 0
\]
Since the discriminant is positive, the equation has two distinct real roots, and \( k = 2 \) is valid.
**Answer:** \( \boxed{2} \)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To determine the value of \( k \) for which one root of the equation \( 2x^2 + 5x + k = 0 \) is the reciprocal of the other, we start by letting the roots be \( r \) and \( \frac{1}{r} \). According to Vieta's formulas, the sum and product of the roots can be expressed as: 1. The sum of the roots \( r + \frac{1}{r} = -\frac{b}{a} = -\frac{5}{2} \). 2. The product of the roots \( r \cdot \frac{1}{r} = 1 = \frac{c}{a} = \frac{k}{2} \). From the second equation, we have: \[ k = 2 \cdot 1 = 2. \] Thus, \( k \) is \( 2 \). Therefore, the value of \( k \) is \( \boxed{2} \).
