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Teacher Guide: Solving Equations by Factoring

Learn how to solve quadratic equations by factoring them into products of binomials and applying the zero product property.

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For Teachers

Learning Objectives
  • Apply the zero product property to solve equations
  • Solve quadratic equations by factoring trinomials
  • Solve equations involving difference of squares
  • Recognize when to factor out a GCF before applying other methods
  • Verify solutions by substitution
Prerequisites
  • • Factoring trinomials (x2+bx+cx^2 + bx + c)
  • • Factoring with leading coefficient (ax2+bx+cax^2 + bx + c)
  • • Difference of squares pattern
  • • Greatest common factor (GCF)
Discussion Starters
  • 1. Why do we call the x-intercepts 'zeros' or 'roots' of an equation?
  • 2. When would factoring be faster than using the quadratic formula?
  • 3. Can you think of a real-life situation where you'd need to find when something equals zero?
  • 4. What happens graphically when a quadratic has no real solutions?
Common Misconceptions

Thinking (x−3)(x−5)=0(x - 3)(x - 5) = 0 means x=−3x = -3 and x=−5x = -5

Remediation: Emphasize: solve each factor by setting it equal to zero. If x−3=0x - 3 = 0, ADD 3 to get x=3x = 3. The solution has the OPPOSITE sign of what appears in the factor.

Assuming all quadratics can be factored

Remediation: Show examples like x2+x+1=0x^2 + x + 1 = 0 which has no real factors. Discuss that the quadratic formula is the backup method.

Differentiation Ideas

For Struggling Students:

  • • Start with equations already in factored form: (x−2)(x+5)=0(x - 2)(x + 5) = 0
  • • Use color-coding: one color for each factor
  • • Provide a checklist: 1) Standard form? 2) GCF? 3) Factor 4) Set each = 0 5) Solve

For On-Level Students:

  • • Mix different factoring types (GCF, trinomial, difference of squares)
  • • Include equations not initially in standard form
  • • Add verification step to all solutions

For Advanced Students:

  • • Equations with higher degree: x3−4x=0x^3 - 4x = 0
  • • Systems involving quadratics
  • • Create equations with specific solutions
Standards Alignment
  • A.REI.B.4b (CCSS.MATH.CONTENT.HSA.REI.B.4.B)

    Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring

  • A.SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)

    Use the structure of an expression to identify ways to rewrite it

Lesson Resources
  • visualSolution Finder

    Graph quadratics and see where they cross the x-axis

  • activityFactor Race

    Timed practice factoring and solving equations

  • worksheetReal-World Applications

    Word problems requiring solving by factoring

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Solving equations by factoring uses the zero product property: if a⋅b=0a \cdot b = 0, then either a=0a = 0 or b=0b = 0 (or both).
To solve a quadratic equation by factoring:
  1. 1.Write the equation in standard form: ax2+bx+c=0ax^2 + bx + c = 0
  2. 2.Factor the left side into a product of binomials
  3. 3.Set each factor equal to zero
  4. 4.Solve each resulting equation
Example:
x2−5x+6=0x^2 - 5x + 6 = 0
(x−2)(x−3)=0(x - 2)(x - 3) = 0
x−2=0 or x−3=0x - 2 = 0 \text{ or } x - 3 = 0
x=2 or x=3x = 2 \text{ or } x = 3

Worked Examples

Solve: x2+7x+12=0x^2 + 7x + 12 = 0
1

Identify the equation form

Already in standard form: x2+7x+12=0x^2 + 7x + 12 = 0 → a=1a = 1, b=7b = 7, c=12c = 12
2

Factor the trinomial

Find two numbers that multiply to 12 and add to 7: 3×4=123 \times 4 = 12 and 3+4=73 + 4 = 7 → (x+3)(x+4)=0(x + 3)(x + 4) = 0
3

Apply zero product property

If (x+3)(x+4)=0(x + 3)(x + 4) = 0, then x+3=0x + 3 = 0 or x+4=0x + 4 = 0 → Two equations to solve
4

Solve each equation

x+3=0⇒x=−3x + 3 = 0 \Rightarrow x = -3 and x+4=0⇒x=−4x + 4 = 0 \Rightarrow x = -4 → x=−3x = -3 or x=−4x = -4

Common Mistakes

Dividing both sides by xx (losing the solution x=0x = 0)

Why it's wrong: When you divide by xx, you assume x≠0x \neq 0. But x=0x = 0 might be a valid solution!

Correct: Factor out xx instead: x2−5x=0⇒x(x−5)=0x^2 - 5x = 0 \Rightarrow x(x - 5) = 0, giving x=0x = 0 or x=5x = 5

Forgetting to set the equation equal to zero first

Why it's wrong: The zero product property only works when the product equals zero, not any other number.

Correct: Always rewrite as expression=0\text{expression} = 0 before factoring. For example: x2=9x^2 = 9 becomes x2−9=0x^2 - 9 = 0

Writing only one solution when there are two

Why it's wrong: Quadratic equations can have two distinct solutions, and both are valid.

Correct: Always check both factors. (x−2)(x−3)=0(x - 2)(x - 3) = 0 gives TWO solutions: x=2x = 2 AND x=3x = 3

Sign errors when factoring

Why it's wrong: The signs in the factored form determine the signs of the solutions.

Correct: (x+3)=0(x + 3) = 0 gives x=−3x = -3 (opposite sign). (x−3)=0(x - 3) = 0 gives x=3x = 3 (same sign)

Why It Matters

Solving equations by factoring is one of the most elegant methods in algebra because it transforms complex problems into simpler ones:
  • Physics: Projectile motion equations (when does the ball hit the ground?)
  • Business: Break-even analysis (when does profit equal zero?)
  • Engineering: Finding dimensions that satisfy area requirements
  • Architecture: Calculating measurements for structural designs
While the quadratic formula always works, factoring is often faster and gives more insight into the structure of the equation!

Real World Applications

Projectile Motion

When an object is thrown upward, its height follows a quadratic equation. Finding when it hits the ground means solving for when height equals zero.

Example:

A ball's height is h=−t2+4t+5h = -t^2 + 4t + 5 meters after tt seconds. When does it hit the ground? Solve −t2+4t+5=0-t^2 + 4t + 5 = 0, or equivalently t2−4t−5=0t^2 - 4t - 5 = 0. Factoring: (t−5)(t+1)=0(t - 5)(t + 1) = 0, so t=5t = 5 seconds (we ignore t=−1t = -1 since time cannot be negative).

1Try It Yourself

A rocket's height is given by h=−2t2+12th = -2t^2 + 12t meters, where tt is time in seconds.

When does the rocket return to the ground?

Step 1: Write the mathematical expression

Set height equal to zero and solve:

Area Problems

Finding dimensions when you know the area often leads to quadratic equations.

Example:

A rectangular garden has length 3 meters more than its width. If the area is 40 square meters, find the dimensions. Let width = xx. Then: x(x+3)=40x(x + 3) = 40, so x2+3x−40=0x^2 + 3x - 40 = 0. Factoring: (x+8)(x−5)=0(x + 8)(x - 5) = 0, giving x=5x = 5 meters (width). Length = 8 meters.

2Try It Yourself

A photo frame has width xx cm and length (x+4)(x + 4) cm. The area of the frame is 45 square cm.

What are the dimensions of the frame?

Step 1: Write the mathematical expression

Set up and solve the area equation:

Key Takeaways

  • 1The zero product property states: if ab=0ab = 0, then a=0a = 0 or b=0b = 0
  • 2Always write the equation in standard form (ax2+bx+c=0ax^2 + bx + c = 0) before factoring
  • 3Factor the expression, then set each factor equal to zero
  • 4Quadratic equations can have zero, one, or two solutions
  • 5Check your answers by substituting back into the original equation
  • 6Never divide by a variable - you might lose solutions!

Frequently Asked Questions

What if the equation doesn't factor nicely?

Not all quadratic equations can be factored with integers. In those cases, use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Factoring works best when the solutions are rational numbers.

Can a quadratic equation have just one solution?

Yes! This happens when the trinomial is a perfect square. For example, x2−6x+9=0x^2 - 6x + 9 = 0 factors as (x−3)2=0(x - 3)^2 = 0, giving only x=3x = 3 (a repeated root).

Why do we set factors equal to zero?

Zero is special: it's the only number where if a product equals it, at least one factor must equal it. If ab=6ab = 6, there are infinite possibilities. But if ab=0ab = 0, either a=0a = 0 or b=0b = 0.

Glossary

Zero Product Property
If a⋅b=0a \cdot b = 0, then a=0a = 0 or b=0b = 0 (or both)
Standard Form
A quadratic equation written as ax2+bx+c=0ax^2 + bx + c = 0
Root/Solution
A value of xx that makes the equation true
Factoring
Writing an expression as a product of simpler expressions

More in This Topic

Introduction to Factoring

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Factoring Out the GCF

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Factoring by Grouping

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Factoring Trinomials (a=1)

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