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: Converting Quadratics to Factored Form**
**Instructions:**
Convert the quadratic into factored form, explaining the steps to change forms. State the roots and vertex.
**Quadratic Expression:**
\[ y = 3x^2 + 6x - 24 \]
**Steps to Convert to Factored Form:**
1. **Identify the Quadratic Coefficients:**
- \( a = 3 \)
- \( b = 6 \)
- \( c = -24 \)
2. **Factor by Grouping:**
- First, factor out the greatest common factor if necessary. In this case, a common factor for the entire expression is 3, so factor that out:
\[ y = 3(x^2 + 2x - 8) \]
3. **Find Pair of Numbers:**
- Look for two numbers that multiply to give the product of the coefficient of \( x^2 \) (which is 1 in the factored quadratic) and the constant term (-8), and add to give the coefficient of x (2).
- The numbers are 4 and -2 because \( 4 \times (-2) = -8 \) and \( 4 + (-2) = 2 \).
4. **Factor the Quadratic:**
- Rewrite the middle term (2x) using the two numbers found:
\[ x^2 + 4x - 2x - 8 \]
- Group the terms:
\[ (x^2 + 4x) - (2x + 8) \]
- Factor each group:
\[ x(x + 4) - 2(x + 4) \]
- Combine with the common factor:
\[ (x - 2)(x + 4) \]
- Remember to multiply back by 3:
\[ y = 3(x - 2)(x + 4) \]
**State the Roots:**
- Set each factor equal to zero:
- \( x - 2 = 0 \rightarrow x = 2 \)
- \( x + 4 = 0 \rightarrow x = -4 \)
**State the Vertex:**
- Use the vertex formula, \( x = -\frac{b}{2a} \):
- With \( a =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F532d2df1-6bef-42be-bbaa-e1703c54e78f%2Fd8ae5e93-9df7-40a9-a352-002eed0f3c88%2Fnx3zzgw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Converting Quadratics to Factored Form**
**Instructions:**
Convert the quadratic into factored form, explaining the steps to change forms. State the roots and vertex.
**Quadratic Expression:**
\[ y = 3x^2 + 6x - 24 \]
**Steps to Convert to Factored Form:**
1. **Identify the Quadratic Coefficients:**
- \( a = 3 \)
- \( b = 6 \)
- \( c = -24 \)
2. **Factor by Grouping:**
- First, factor out the greatest common factor if necessary. In this case, a common factor for the entire expression is 3, so factor that out:
\[ y = 3(x^2 + 2x - 8) \]
3. **Find Pair of Numbers:**
- Look for two numbers that multiply to give the product of the coefficient of \( x^2 \) (which is 1 in the factored quadratic) and the constant term (-8), and add to give the coefficient of x (2).
- The numbers are 4 and -2 because \( 4 \times (-2) = -8 \) and \( 4 + (-2) = 2 \).
4. **Factor the Quadratic:**
- Rewrite the middle term (2x) using the two numbers found:
\[ x^2 + 4x - 2x - 8 \]
- Group the terms:
\[ (x^2 + 4x) - (2x + 8) \]
- Factor each group:
\[ x(x + 4) - 2(x + 4) \]
- Combine with the common factor:
\[ (x - 2)(x + 4) \]
- Remember to multiply back by 3:
\[ y = 3(x - 2)(x + 4) \]
**State the Roots:**
- Set each factor equal to zero:
- \( x - 2 = 0 \rightarrow x = 2 \)
- \( x + 4 = 0 \rightarrow x = -4 \)
**State the Vertex:**
- Use the vertex formula, \( x = -\frac{b}{2a} \):
- With \( a =
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