Quadratic Factoring Worksheet
Miami-Dade & Florida Mathematics Curriculum • Secondary Mathematics
Factoring & Solving Quadratics
ALGEBRA 2 & HONORS
<table class="w-full text-xs border border-slate-300 mb-2 border-collapse"><tbody><tr class="bg-slate-50"><td class="px-2 py-1 border border-slate-300 font-semibold text-slate-700 w-1/2">Student Name:<span class="inline-block w-44 border-b border-dotted border-slate-400 ml-1"></span></td><td class="px-2 py-1 border border-slate-300 font-semibold text-slate-700 w-1/4">Date:<span class="inline-block w-20 border-b border-dotted border-slate-400 ml-1"></span></td><td class="px-2 py-1 border border-slate-300 font-semibold text-slate-700 w-1/4">Period:<span class="inline-block w-14 border-b border-dotted border-slate-400 ml-1"></span></td></tr></tbody></table>
Florida B.E.S.T. Alignment: MA.912.AR.3.1 (Solve quadratics by factoring) & MA.912.AR.1.3 (Factor polynomials completely).
TOTAL: 56 PTS
Part 1: Factoring Monic Trinomials and Greatest Common Factors (GCF)
Factor completely over the integers.
1. Basic Trinomial (\(a=1\)) [3 pts]
\(x^2 - 11x + 24\)
Factored Form:
2. GCF Factorization First [3 pts]
\(3x^3 + 6x^2 - 45x\)
Factored Form:
3. Difference of Two Squares [3 pts]
\(49x^2 - 36\)
Factored Form:
4. Perfect Square Trinomial [3 pts]
\(4x^2 - 28x + 49\)
Factored Form:
Part 2: Advanced Factoring (\(a > 1\)) and Factoring by Grouping
Show method (ac-method, grouping, or box).
5. Trinomial with \(a > 1\) [4 pts]
\(2x^2 + 7x - 15\)
Factored Form:
6. Trinomial with Negative \(c\) [4 pts]
\(6x^2 - 11x - 10\)
Factored Form:
7. Factoring by Grouping (4 Terms) [4 pts]
\(3x^3 + 2x^2 - 12x - 8\)
Factored Form:
8. GCF with Negative Coefficient [4 pts]
\(-4x^2 - 20x + 96\)
Factored Form:
Algebra 2 & Honors • Unit 3: Quadratic Functions and Equations Florida B.E.S.T. MA.912.AR.3.1 Page 1 of 2
Algebra 2 & Honors: Solving Quadratics | Student:
Period: Page 2 of 2
Part 3: Solving Quadratic Equations by Factoring (Zero Product Property)
Rewrite as \(ax^2 + bx + c = 0\) before factoring.
9. Standard Monic Equation [4 pts]
\(x^2 - 3x - 28 = 0\)
Solution Set:
\(x = \)
10. Rearrangement Required [4 pts]
\(x^2 + 14x = -49\)
Solution Set:
\(x = \)
11. Non-Monic Rearrangement [5 pts]
\(5x^2 - 13x = 6\)
Solution Set:
\(x = \)
12. Binomial Missing Constant [4 pts]
\(8x^2 = 20x\)
Solution Set:
\(x = \)
Part 4: Honors Application & Higher-Order Modeling
FL B.E.S.T. MA.912.AR.3.1 Extension
13. [HONORS] Equations Quadratic in Form [5 pts]
Solve the quartic polynomial equation over the real number system by factoring as a quadratic in form: \(2x^4 - 7x^2 - 4 = 0\). State all real solutions.
Real Solutions:
\(x = \)
14. [APPLICATION] Geometric Area Modeling [6 pts]
A rectangular community courtyard in Orlando measures \(12\text{ m}\) by \(16\text{ m}\). A paved pedestrian walkway of uniform width \(w\) is constructed entirely around the exterior border of the courtyard. The total combined area of the courtyard and walkway is \(320\text{ m}^2\).
(a) Write a quadratic equation in terms of \(w\). (b) Solve by factoring to find the walkway width \(w\).
Standard Equation:
Walkway Width \(w\):
meters
Algebra 2 & Honors • Unit 3: Quadratic Functions and Equations Florida B.E.S.T. MA.912.AR.3.1 Page 2 of 2
Quadratic Factoring Key
TEACHER RESOURCE Florida Mathematics Curriculum • Algebra 2
Answer Key & Worked Solutions
COMPLETE KEY • 56 PTS
<table class="w-full text-xs border border-slate-300 mb-2 border-collapse"><tbody><tr class="bg-slate-50"><td class="px-2.5 py-1 border border-slate-300 text-slate-700 w-2/3"><strong class="text-emerald-900 font-bold">FL B.E.S.T. Standards:</strong> <span class="font-semibold text-slate-900">MA.912.AR.3.1</span> & <span class="font-semibold text-slate-900">MA.912.AR.1.3</span></td><td class="px-2.5 py-1 border border-slate-300 text-slate-700 w-1/3 text-right"><span class="font-bold text-slate-800">Distribution:</span> P1: 12p | P2: 16p | P3: 17p | P4: 11p</td></tr></tbody></table>
Part 1 Solutions: Monic Trinomials & GCF (12 Points Total)
3 Points Each
1. \(x^2 - 11x + 24\) [3 pts]
• Identify factor pairs of \(+24\) with sum \(-11\): \((-3) \times (-8) = 24\); \((-3) + (-8) = -11\).
• Split into binomial factors directly since \(a = 1\).
Answer: \(\mathbf{(x - 3)(x - 8)}\)
2. \(3x^3 + 6x^2 - 45x\) [3 pts]
• Factor out greatest common factor \(3x\) first: \(3x(x^2 + 2x - 15)\).
• Trinomial factors: \((+5) \times (-3) = -15\) and \(5 + (-3) = 2\).
Answer: \(\mathbf{3x(x + 5)(x - 3)}\)
3. \(49x^2 - 36\) [3 pts]
• Difference of two squares: \(a^2 - b^2 = (a - b)(a + b)\).
• Here \(a = \sqrt{49x^2} = 7x\) and \(b = \sqrt{36} = 6\).
Answer: \(\mathbf{(7x - 6)(7x + 6)}\)
4. \(4x^2 - 28x + 49\) [3 pts]
• Perfect square trinomial: \(a^2 - 2ab + b^2 = (a - b)^2\).
• Check middle term: \(2(2x)(7) = 28x\). Exact match.
Answer: \(\mathbf{(2x - 7)^2}\)
Part 2 Solutions: Non-Monic (\(a > 1\)) & Grouping (16 Points Total)
4 Points Each
5. \(2x^2 + 7x - 15\) [4 pts]
• \(ac = 2(-15) = -30\). Factors with sum \(+7\): \(10\) and \(-3\).
• Grouping: \(2x^2 + 10x - 3x - 15 = 2x(x + 5) - 3(x + 5)\).
Answer: \(\mathbf{(2x - 3)(x + 5)}\)
6. \(6x^2 - 11x - 10\) [4 pts]
• \(ac = 6(-10) = -60\). Factors with sum \(-11\): \(-15\) and \(+4\).
• Grouping: \(6x^2 - 15x + 4x - 10 = 3x(2x - 5) + 2(2x - 5)\).
Answer: \(\mathbf{(3x + 2)(2x - 5)}\)
7. \(3x^3 + 2x^2 - 12x - 8\) [4 pts]
• Group pairs: \(x^2(3x + 2) - 4(3x + 2) = (x^2 - 4)(3x + 2)\).
• Factor \(x^2 - 4\) completely: \((x - 2)(x + 2)(3x + 2)\).
Answer: \(\mathbf{(x - 2)(x + 2)(3x + 2)}\)
8. \(-4x^2 - 20x + 96\) [4 pts]
• Factor out \(-4\) first: \(-4(x^2 + 5x - 24)\).
