Algebra 1 Practice - Factor a Quadratic Expression Where Leading Coefficient is Not 1 (Example 1)

In **Algebra 1**, factoring a quadratic expression involves rewriting it as a product of two binomials. This is a key skill used to solve quadratic equations and simplify expressions.

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### **Standard Form of a Quadratic Expression**
\[
ax^2 + bx + c
\]
Where:
- \( a \) is the coefficient of \( x^2 \),
- \( b \) is the coefficient of \( x \),
- \( c \) is the constant term.

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### **Steps to Factor a Quadratic Expression**

1. **Check for a Greatest Common Factor (GCF):**
- If all terms share a common factor, factor it out first.

2. **Identify the Coefficients:**
- Note \( a \), \( b \), and \( c \) from the quadratic expression.

3. **Choose a Factoring Method:**
- **If \( a = 1 \):** Find two numbers that multiply to \( c \) and add to \( b \).
Write the factors as \( (x + m)(x + n) \).
- **If \( a \neq 1 \):** Use one of these methods:
- **Trial and Error:** Guess and test pairs of factors.
- **Split the Middle Term:** Rewrite \( bx \) as two terms and factor by grouping.

4. **Double-Check:**
Expand the factors to ensure they produce the original quadratic.

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### **Examples**

1. **Factor \( x^2 + 7x + 10 \):**
- \( a = 1 \), \( b = 7 \), \( c = 10 \).
- Find two numbers that multiply to \( 10 \) and add to \( 7 \): \( 5 \) and \( 2 \).
- Write as \( (x + 5)(x + 2) \).
- **Answer**: \( (x + 5)(x + 2) \).

2. **Factor \( x^2 - 5x + 6 \):**
- \( a = 1 \), \( b = -5 \), \( c = 6 \).
- Find two numbers that multiply to \( 6 \) and add to \( -5 \): \( -3 \) and \( -2 \).
- Write as \( (x - 3)(x - 2) \).
- **Answer**: \( (x - 3)(x - 2) \).

3. **Factor \( 2x^2 + 5x + 3 \):**
- \( a = 2 \), \( b = 5 \), \( c = 3 \).
- Multiply \( a \) and \( c \): \( 2 \cdot 3 = 6 \).
- Find two numbers that multiply to \( 6 \) and add to \( 5 \): \( 3 \) and \( 2 \).
- Rewrite as \( 2x^2 + 3x + 2x + 3 \).
- Factor by grouping: \( x(2x + 3) + 1(2x + 3) = (x + 1)(2x + 3) \).
- **Answer**: \( (x + 1)(2x + 3) \).

4. **Factor \( 3x^2 - 4x - 7 \):**
- \( a = 3 \), \( b = -4 \), \( c = -7 \).
- Multiply \( a \) and \( c \): \( 3 \cdot -7 = -21 \).
- Find two numbers that multiply to \( -21 \) and add to \( -4 \): \( -7 \) and \( 3 \).
- Rewrite as \( 3x^2 - 7x + 3x - 7 \).
- Factor by grouping: \( x(3x - 7) + 1(3x - 7) = (x + 1)(3x - 7) \).
- **Answer**: \( (x + 1)(3x - 7) \).

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### **Practice Problems**

1. Factor \( x^2 + 8x + 12 \).
2. Factor \( x^2 - 6x + 9 \).
3. Factor \( 2x^2 + 7x + 3 \).
4. Factor \( 3x^2 - 5x - 2 \).
5. Factor \( x^2 - 10x + 21 \).

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### **Tips**
- Always check for a GCF before factoring.
- Practice identifying pairs of numbers that satisfy the conditions quickly.
- Expand your answer to verify correctness.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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