Factor Forge Worksheet
Directions: Solve each equation by factoring. Clearly show all steps and state all real solutions.
Name:
Date:
Period:
31 \( x^2 - 7x = 18 \)
Trinomial
Solution(s):
\( x = \)
32 \( 2x^2 + 13x + 15 = 0 \)
\( a > 1 \)
Solution(s):
\( x = \)
33 \( 3x^2 - 12x = 36 \)
GCF • Trinomial
Solution(s):
\( x = \)
34 \( 6x^2 - 7x - 5 = 0 \)
AC Method
Solution(s):
\( x = \)
35 \( 4x^3 - 36x = 0 \)
Cubic • Squares
Solution(s):
\( x = \)
36 \( x^3 - 4x^2 - 9x + 36 = 0 \)
4-Term Grouping
Solution(s):
\( x = \)
Checklist: 1. Set to 0 • 2. Factor out GCF • 3. Factor completely • 4. Zero Product Property • 5. Solve for \( x \)
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Answer Key Problems #31 – #36 Solutions
Step-by-Step Factoring Guide
31 \( x^2 - 7x = 18 \)
Standard Trinomial
1. Standard Form: \( x^2 - 7x - 18 = 0 \)
2. Factor Pair: Factors of \( -18 \) summing to \( -7 \) are \( -9 \) and \( 2 \).
3. Factored Form: \( (x - 9)(x + 2) = 0 \)
4. Zero Product: \( x - 9 = 0 \implies x = 9 \); \( x + 2 = 0 \implies x = -2 \)
Final Solution: \( x = 9, \; -2 \)
32 \( 2x^2 + 13x + 15 = 0 \)
Leading Coeff \( a > 1 \)
1. Split Term: \( ac = 2(15) = 30 \). Factors adding to \( 13 \) are \( 10 \) and \( 3 \).
2. Rewrite: \( 2x^2 + 10x + 3x + 15 = 0 \)
3. Group: \( 2x(x + 5) + 3(x + 5) = (2x + 3)(x + 5) = 0 \)
4. Zero Product: \( 2x + 3 = 0 \implies x = -\frac{3}{2} \); \( x + 5 = 0 \implies x = -5 \)
Final Solution: \( x = -\frac{3}{2}, \; -5 \)
33 \( 3x^2 - 12x = 36 \)
GCF • Quadratic
1. Standard Form: \( 3x^2 - 12x - 36 = 0 \)
2. Extract GCF: \( 3(x^2 - 4x - 12) = 0 \)
3. Factor Trinomial: Factors of \( -12 \) summing to \( -4 \) are \( -6 \) and \( 2 \): \( 3(x - 6)(x + 2) = 0 \)
4. Solve: \( x - 6 = 0 \implies x = 6 \); \( x + 2 = 0 \implies x = -2 \)
Final Solution: \( x = 6, \; -2 \)
34 \( 6x^2 - 7x - 5 = 0 \)
AC Factoring Method
1. Find Pair: \( ac = 6(-5) = -30 \). Factors summing to \( -7 \) are \( -10 \) and \( 3 \).
2. Split: \( 6x^2 - 10x + 3x - 5 = 0 \)
3. Group: \( 2x(3x - 5) + 1(3x - 5) = (2x + 1)(3x - 5) = 0 \)
4. Solve: \( 2x + 1 = 0 \implies x = -\frac{1}{2} \); \( 3x - 5 = 0 \implies x = \frac{5}{3} \)
Final Solution: \( x = -\frac{1}{2}, \; \frac{5}{3} \)
35 \( 4x^3 - 36x = 0 \)
Cubic • Diff of Squares
1. Monomial GCF: Factor out \( 4x \): \( 4x(x^2 - 9) = 0 \)
2. Diff of Squares: \( x^2 - 9 = (x - 3)(x + 3) \)
3. Complete Factorization: \( 4x(x - 3)(x + 3) = 0 \)
4. Solve (3 roots): \( 4x = 0 \implies x = 0 \); \( x - 3 = 0 \implies x = 3 \); \( x + 3 = 0 \implies x = -3 \)
Final Solution: \( x = 0, \; 3, \; -3 \)
36 \( x^3 - 4x^2 - 9x + 36 = 0 \)
4-Term Grouping
1. Group Pairs: \( (x^3 - 4x^2) - (9x - 36) = 0 \)
2. Factor Binomial: \( x^2(x - 4) - 9(x - 4) = 0 \implies (x^2 - 9)(x - 4) = 0 \)
3. Decompose Difference: \( (x - 3)(x + 3)(x - 4) = 0 \)
4. Solve (3 roots): \( x = 3, \; x = -3, \; x = 4 \)
Final Solution: \( x = 3, \; -3, \; 4 \)
Teacher Note: Remind students not to divide both sides by variable factors, and to check each solution in the original equation.
Page 2 of 2 (Solutions)
