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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![**Title: Solving Quadratic Equations by Factoring**
In this educational note, we will solve a quadratic equation using the factoring method.
The given equation is:
\[ x^2 - x - 30 = 0 \]
**Step-by-Step Solution:**
1. **Identify the quadratic equation:** The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, \(a = 1\), \(b = -1\), and \(c = -30\).
2. **Factor the quadratic expression:** Look for two numbers that multiply to give \( ac = 1 \times -30 = -30 \) and add to give \( b = -1 \).
The pair of numbers that fit this criterion are \( 5 \) and \( -6 \), since \( 5 \times -6 = -30 \) and \( 5 + (-6) = -1 \).
3. **Rewrite the middle term using the identified numbers:**
\[ x^2 + 5x - 6x - 30 = 0 \]
4. **Group the terms and factor by grouping:**
\[ (x^2 + 5x) + (-6x - 30) = 0 \]
\[ x(x + 5) - 6(x + 5) = 0\]
5. **Factor out the common binomial factor:**
\[ (x - 6)(x + 5) = 0 \]
6. **Solve for \( x \):**
Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 6 \quad \text{or} \quad x = -5
\]
Thus, the solutions to the quadratic equation \( x^2 - x - 30 = 0 \) are \( x = 6 \) and \( x = -5 \).
Graph and Diagram Explanation:
- The given handwritten notes contain a few other markings and calculations, but the primary focus is on the quadratic equation \( x^2 - x - 30 = 0 \).
- It appears there might be some partial solution attempts, possibly leading to the correct factorization mentioned above.
- There is also a fraction \( \frac{12}{30} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2Fdf08834d-64c1-41bb-b15e-1b05e23bbc08%2Fz0mylan_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Quadratic Equations by Factoring**
In this educational note, we will solve a quadratic equation using the factoring method.
The given equation is:
\[ x^2 - x - 30 = 0 \]
**Step-by-Step Solution:**
1. **Identify the quadratic equation:** The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, \(a = 1\), \(b = -1\), and \(c = -30\).
2. **Factor the quadratic expression:** Look for two numbers that multiply to give \( ac = 1 \times -30 = -30 \) and add to give \( b = -1 \).
The pair of numbers that fit this criterion are \( 5 \) and \( -6 \), since \( 5 \times -6 = -30 \) and \( 5 + (-6) = -1 \).
3. **Rewrite the middle term using the identified numbers:**
\[ x^2 + 5x - 6x - 30 = 0 \]
4. **Group the terms and factor by grouping:**
\[ (x^2 + 5x) + (-6x - 30) = 0 \]
\[ x(x + 5) - 6(x + 5) = 0\]
5. **Factor out the common binomial factor:**
\[ (x - 6)(x + 5) = 0 \]
6. **Solve for \( x \):**
Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 6 \quad \text{or} \quad x = -5
\]
Thus, the solutions to the quadratic equation \( x^2 - x - 30 = 0 \) are \( x = 6 \) and \( x = -5 \).
Graph and Diagram Explanation:
- The given handwritten notes contain a few other markings and calculations, but the primary focus is on the quadratic equation \( x^2 - x - 30 = 0 \).
- It appears there might be some partial solution attempts, possibly leading to the correct factorization mentioned above.
- There is also a fraction \( \frac{12}{30} \)
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