Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Problem Statement:
Given the quadratic equation:
\[ 3x^2 + 36x + 121 = 13 \]
**Task:** Determine the number of real solutions.
### Explanation:
To find the number of real solutions for a quadratic equation of the form \( ax^2 + bx + c = 0 \), we use the discriminant, \( \Delta \), where:
\[ \Delta = b^2 - 4ac \]
1. **Rearrange the equation** to standard form:
\[ 3x^2 + 36x + 108 = 0 \]
(by subtracting 13 from both sides)
2. **Identify the coefficients**:
- \( a = 3 \)
- \( b = 36 \)
- \( c = 108 \)
3. **Calculate the discriminant**:
\[ \Delta = 36^2 - 4 \times 3 \times 108 \]
4. **Determine the number of solutions**:
- If \( \Delta > 0 \), there are 2 distinct real solutions.
- If \( \Delta = 0 \), there is 1 real solution (repeated).
- If \( \Delta < 0 \), there are no real solutions (the solutions are complex).
Ensure the calculations are carried out correctly to confirm the number of real solutions for this equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d59e81a-3610-4fe8-95bd-8152142fc5c0%2F93a93560-406d-4d25-bfd1-de87a1e8b1af%2Fsho9ywc.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Given the quadratic equation:
\[ 3x^2 + 36x + 121 = 13 \]
**Task:** Determine the number of real solutions.
### Explanation:
To find the number of real solutions for a quadratic equation of the form \( ax^2 + bx + c = 0 \), we use the discriminant, \( \Delta \), where:
\[ \Delta = b^2 - 4ac \]
1. **Rearrange the equation** to standard form:
\[ 3x^2 + 36x + 108 = 0 \]
(by subtracting 13 from both sides)
2. **Identify the coefficients**:
- \( a = 3 \)
- \( b = 36 \)
- \( c = 108 \)
3. **Calculate the discriminant**:
\[ \Delta = 36^2 - 4 \times 3 \times 108 \]
4. **Determine the number of solutions**:
- If \( \Delta > 0 \), there are 2 distinct real solutions.
- If \( \Delta = 0 \), there is 1 real solution (repeated).
- If \( \Delta < 0 \), there are no real solutions (the solutions are complex).
Ensure the calculations are carried out correctly to confirm the number of real solutions for this equation.
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