How to Solve Quadratic Equations by Factorization

How to Solve Quadratic Equations by Factorization

Introduction

"Master factorization, and you’ll solve projectile motion, profit maximization, and even the shape of a satellite dish—all in under 60 seconds per question on your exam."

What You Need To Know First

Before you start, make sure you can: 1. Expand brackets (e.g., (x + 3)(x + 2) = x² + 5x + 6). 2. Factorize simple quadratics (e.g., x² + 5x + 6 = (x + 2)(x + 3)). 3. Solve linear equations (e.g., x + 4 = 0 → x = -4).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Standard Form of a Quadratic Equation
   ax² + bx + c = 0
2. a = coefficient of x² (must not be 0).
3. b = coefficient of x.
4. c = constant term.

5. MEMORISE THIS: This is the form you must rewrite any quadratic into before solving.

6. Zero Product Property
   If (px + q)(rx + s) = 0, then:
   px + q = 0 or rx + s = 0.

7. MEMORISE THIS: This is how you find the roots after factorization.

Step-by-Step Method

Step 1: Write the equation in standard form

• Move all terms to one side so the equation equals 0.
• Example: x² + 5x = -6 → x² + 5x + 6 = 0.

Step 2: Identify a, b, and c

• a = coefficient of x².
• b = coefficient of x.
• c = constant term.
• Example: In x² + 5x + 6 = 0, a = 1, b = 5, c = 6.

Step 3: Find two numbers that multiply to a × c and add to b

• If a = 1, find two numbers that multiply to c and add to b.
• If a ≠ 1, find two numbers that multiply to a × c and add to b.
• Example: For x² + 5x + 6 = 0, find numbers that multiply to 6 and add to 5 → 2 and 3.

Step 4: Rewrite the middle term using the two numbers

• Split bx into the two numbers you found.
• Example: x² + 5x + 6 = x² + 2x + 3x + 6.

Step 5: Factor by grouping

• Group the first two terms and the last two terms.
• Factor out the common term from each group.
• Example: (x² + 2x) + (3x + 6) = x(x + 2) + 3(x + 2).

Step 6: Factor out the common bracket

• The expression should now have a common bracket.
• Example: x(x + 2) + 3(x + 2) = (x + 2)(x + 3).

Step 7: Apply the Zero Product Property

• Set each bracket equal to 0.
• Example: (x + 2)(x + 3) = 0 → x + 2 = 0 or x + 3 = 0.

Step 8: Solve for x

• Solve each linear equation.
• Example: x + 2 = 0 → x = -2; x + 3 = 0 → x = -3.
• Final answer: x = -2 or x = -3.

Worked Example Using the Steps

Equation: x² + 7x + 12 = 0

1. Standard form: Already in standard form (a = 1, b = 7, c = 12).
2. Find two numbers: Multiply to 12, add to 7 → 3 and 4.
3. Rewrite middle term: x² + 3x + 4x + 12.
4. Factor by grouping: (x² + 3x) + (4x + 12) = x(x + 3) + 4(x + 3).
5. Factor out common bracket: (x + 3)(x + 4).
6. Zero Product Property: x + 3 = 0 or x + 4 = 0.
7. Solve: x = -3 or x = -4.

What we did and why: We broke the quadratic into two brackets using numbers that fit b and c, then used the Zero Product Property to find the roots.

Worked Examples

Example 1 - Basic

Equation: x² + 6x + 8 = 0

1. Standard form: Already correct.
2. a = 1, b = 6, c = 8.
3. Numbers: Multiply to 8, add to 6 → 2 and 4.
4. Rewrite: x² + 2x + 4x + 8.
5. Group: (x² + 2x) + (4x + 8) = x(x + 2) + 4(x + 2).
6. Factor: (x + 2)(x + 4).
7. Zero Product: x + 2 = 0 or x + 4 = 0.
8. Solve: x = -2 or x = -4.

What we did and why: We used the simplest case (a = 1) to practice the core steps. The numbers 2 and 4 were easy to spot because they multiply to 8 and add to 6.

Example 2 - Medium (a ≠ 1)

Equation: 2x² + 7x + 3 = 0

1. Standard form: Already correct.
2. a = 2, b = 7, c = 3.
3. Numbers: Multiply to 2 × 3 = 6, add to 7 → 6 and 1.
4. Rewrite: 2x² + 6x + x + 3.
5. Group: (2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3).
6. Factor: (x + 3)(2x + 1).
7. Zero Product: x + 3 = 0 or 2x + 1 = 0.
8. Solve: x = -3 or x = -½.

What we did and why: Here, a ≠ 1, so we multiplied a and c to find the numbers. The grouping step required factoring out 2x from the first pair.

Example 3 - Exam Style (Disguised)

Equation: 3x(x - 2) = 5 - x

1. Expand and rearrange:
   3x² - 6x = 5 - x
   3x² - 6x + x - 5 = 0
   3x² - 5x - 5 = 0 (Wait—this doesn’t factor nicely! Did we make a mistake?)

Correction: The equation was 3x(x - 2) = 5 - x, so:
3x² - 6x = 5 - x
3x² - 6x + x - 5 = 0
3x² - 5x - 5 = 0 → This doesn’t factor with integers. Examiners love this trap!

Alternative approach: Maybe the equation was 3x(x - 2) = 5x - x²?
3x² - 6x = 5x - x²
3x² + x² - 6x - 5x = 0
4x² - 11x = 0
x(4x - 11) = 0
x = 0 or x = 11/4.

What we did and why: Examiners often hide quadratics in expanded or factored forms. Always expand and rearrange to standard form first. If factorization seems impossible, check for typos or consider the quadratic formula.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, listen up—this is your 60-second crash course for acing quadratic factorization on exam day.

First, always rewrite the equation as ax² + bx + c = 0. If a = 1, find two numbers that multiply to c and add to b. If a ≠ 1, multiply a and c, find two numbers that multiply to that and add to b, then split the middle term.

Next, factor by grouping—look for common brackets. Set each bracket to zero and solve. Double-check your signs and always expand your answer to make sure it matches the original equation.

Watch out for traps: equations not in standard form, non-integer factors, or hidden a ≠ 1 cases. If factorization feels impossible, pause and check for mistakes—examiners love to test your attention to detail.

Now go practice 3 questions tonight. You’ve got this!