Factoring Quadratic Expressions

Grade 10 · mathematics · 91 practice problems · read aloud

🔊 Listen to this explanation

🧩 What is Factoring Quadratics?

Factoring is rewriting a quadratic expression (like ax² + bx + c) as a product of two simpler binomials. For example, turning x² + 5x + 6 into (x + 2)(x + 3). This is incredibly useful because it allows us to find the roots (or x-intercepts) of a quadratic equation, which is essential for graphing and solving real-world problems.

📝 Step-by-Step Guide (a = 1)

When the coefficient of x² is 1, follow these steps:

  1. Identify a, b, and c in the trinomial ax² + bx + c.
  2. Find two numbers that:
    • Multiply to equal c
    • Add to equal b
  3. Write the factors as (x + first number)(x + second number).
  4. Check your work by expanding (using FOIL) to ensure you get the original expression.

🔍 Worked Examples

Example 1: Factor x² + 7x + 12

  1. a=1, b=7, c=12
  2. Find two numbers that multiply to 12 and add to 7.
    ✅ Possibilities: 3 & 4 (because 3×4=12 and 3+4=7)
  3. Write the factors: (x + 3)(x + 4)
  4. Check: (x+3)(x+4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✔

Example 2: Factor x² - 3x - 10

  1. a=1, b=-3, c=-10
  2. Find two numbers that multiply to -10 and add to -3.
    ✅ Possibilities: -5 & 2 (because -5×2=-10 and -5+2=-3)
  3. Write the factors: (x - 5)(x + 2)
  4. Check: (x-5)(x+2) = x² + 2x - 5x - 10 = x² - 3x - 10 ✔

⚠️ Common Mistakes to Avoid

  • Incorrect Signs: This is the #1 error! Pay close attention to whether b and c are positive or negative. If c is positive, both numbers have the same sign. If c is negative, the numbers have opposite signs.
  • Forgetting to Factor Out a GCF First: Always look for a Greatest Common Factor in all three terms before you start. For example, in 2x² + 8x + 6, first factor out the 2.
  • Assuming All Quadratics are Factorable: Not all quadratics factor into nice integers. Some require the quadratic formula.

💡 Tips & Tricks

  • The "AC" Method (for a ≠ 1): Multiply a and c. Find two numbers that multiply to this product and add to b. Rewrite the middle term and factor by grouping.
  • Memory Aid: Think "M.A." - Multiply to Add. It reminds you of the relationship between the two numbers you're looking for.
  • Check Your Factors! A quick FOIL check takes seconds and can save you from losing points on a test.

🎯 How to Practice

Mastery comes with consistent practice. Try this sequence:

  1. Start with simple trinomials where a=1 and all terms are positive.
  2. Move to trinomials where a=1 but b and/or c are negative.
  3. Challenge yourself with trinomials where a > 1.
  4. Create your own problems and solve them, or try to factor the quadratics you see in your homework and textbook examples before looking at the solution.

Practice problems

6 of the 91, worked through step by step — try them before opening the answer.

1 x² - 6x + 9 = ?

Hint: Look for a perfect square trinomial pattern where the first and last terms are perfect squares

Show the answer

Answer: (x-3)²

  1. Identify the trinomial: x² - 6x + 9
  2. Check if it's a perfect square: x² is (x)² and 9 is (3)²
  3. Verify the middle term: 2 × x × 3 = 6x, which matches the absolute value of -6x
  4. Since the middle term is negative, write as (x - 3)²
  5. The factored form is (x - 3)²

The answer is (x-3)².

2 x² - 7x + 12 = 0

Hint: Look for two numbers that multiply to the constant term and add to the coefficient of the middle term.

Show the answer

Answer: x = 3, 4

  1. Identify the coefficients. The equation is in the form: a x² + b x + c = 0 Here, a = 1, b = -7, c = 12.
  2. Factor the quadratic expression. We look for two numbers that multiply to a * c = 1 * 12 = 12, and add up to b = -7. List factor pairs of 12: 1 and 12 → sum = 13 (not -7) 2 and 6 → sum = 8 (not -7) 3 and 4 → sum = 7 (not -7) Since we need a negative sum (-7), we check negative factor pairs: -1 and -12 → sum = -13 (not -7) -2 and -6 → sum = -8 (not -7) -3 and -4 → sum = -7 (yes, this works) So the numbers are -3 and -4.
  3. Write the factored form. x² - 7x + 12 = (x - 3)(x - 4) = 0
  4. Apply the zero-product property. If (x - 3)(x - 4) = 0, then at least one factor must be zero. So, x - 3 = 0 or x - 4 = 0
  5. Solve each equation. x - 3 = 0 → x = 3 x - 4 = 0 → x = 4
  6. Final answer. x = 3, 4

We are solving the quadratic equation: x² - 7x + 12 = 0

3 x² - 5x - 14 = 0

Hint: Look for two numbers that multiply to the constant term and add to the coefficient of the middle term

Show the answer

Answer: x = 7, -2

  1. Identify the quadratic equation: x² - 5x - 14 = 0
  2. Find two numbers that multiply to -14 and add to -5
  3. The numbers are -7 and 2 because (-7) × 2 = -14 and (-7) + 2 = -5
  4. Factor the quadratic: (x - 7)(x + 2) = 0
  5. Set each factor equal to zero: x - 7 = 0 or x + 2 = 0
  6. Solve each equation: x = 7 or x = -2 The solutions are x = 7 and x = -2.

4 x² - 5x - 36 = 0

Hint: Look for two numbers that multiply to the constant term and add to the coefficient of the x-term

Show the answer

Answer: x = 9, -4

  1. Identify the quadratic equation: x² - 5x - 36 = 0
  2. Find two numbers that multiply to -36 and add to -5
  3. The numbers are -9 and 4 because (-9) × 4 = -36 and (-9) + 4 = -5
  4. Factor the quadratic: (x - 9)(x + 4) = 0
  5. Set each factor equal to zero: x - 9 = 0 or x + 4 = 0
  6. Solve each equation: x = 9 or x = -4 The solutions are x = 9 and x = -4.

5 2x² + 7x - 4 = 0

Hint: Look for two numbers that multiply to give the product of the coefficient of x² and the constant term, and add to give the coefficient of x.

Show the answer

Answer: x = 0.5, x = -4

  1. Identify a = 2, b = 7, c = -4
  2. Multiply a and c: 2 × (-4) = -8
  3. Find two numbers that multiply to -8 and add to 7: 8 and -1
  4. Rewrite the middle term: 2x² + 8x - x - 4 = 0
  5. Factor by grouping: (2x² + 8x) + (-x - 4) = 0
  6. Factor out common terms: 2x(x + 4) - 1(x + 4) = 0
  7. Factor out (x + 4): (2x - 1)(x + 4) = 0
  8. Set each factor equal to zero: 2x - 1 = 0 or x + 4 = 0
  9. Solve each equation: x = 1/2 = 0.5 or x = -4 The solutions are x = 0.5 and x = -4.

6 Factor: 9x² - 49

Hint: This is a difference of squares. Look for two perfect squares separated by a minus sign.

Show the answer

Answer: (3x - 7)(3x + 7)

  1. Recognize the expression as a difference of squares: 9x² - 49 = (3x)² - 7².
  2. Apply the formula a² - b² = (a - b)(a + b), where a = 3x and b = 7.
  3. Write the factors: (3x - 7)(3x + 7).

The answer is (3x - 7)(3x + 7).

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