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Factoring Simple Quadratics

A free Algebra II lesson from the “Polynomial Factorization” unit, with a worked example and practice problems including step-by-step solutions.

Factoring quadratics rewrites a trinomial as a product of two binomials. This is useful because a product is easier to solve when it equals zero: if one factor is zero, the whole product is zero. For expressions like x^2 + bx + c, look for two numbers that multiply to c and add to b. When the leading coefficient is not 1, also track how the first terms multiply to the x^2 term. A common mistake is finding numbers that multiply correctly but do not add to the middle coefficient. Always multiply the factors back to check.

What you'll learn

Why it matters: Profit equations, projectile zero-crossings, and area-product problems factor into two binomials. Once an expression is written as (x − a)(x − b), the zero-product property hands you the solutions almost for free.

Worked example

Problem. Factor x^2 + 7x + 10.

  1. Find two numbers that multiply to 10.
  2. 5 and 2 multiply to 10 and add to 7.
  3. Write the factors as (x + 5)(x + 2).

Answer: (x + 5)(x + 2)

Practice problems

1. Factor x^2 + 5x + 6.

Choices: (x + 2)(x + 3) · (x + 1)(x + 6) · (x - 2)(x - 3) · (x + 5)(x + 1)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Factor x^2 + 5x + 6.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 2 and 3 multiply to 6.
  4. 2 and 3 add to 5.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x + 2)(x + 3)

2. For x^2 - 9x + 20, what two positive numbers multiply to 20 and add to 9? Enter the larger one.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For x^2 - 9x + 20, what two positive numbers multiply to 20 and add to 9? Enter the larger one.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Factor pairs of 20 include 1 and 20, 2 and 10, 4 and 5.
  4. 4 + 5 = 9.
  5. The larger number is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

3. Factor x^2 - x - 12.

Choices: (x - 4)(x + 3) · (x + 4)(x - 3) · (x - 6)(x + 2) · (x - 12)(x + 1)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Factor x^2 - x - 12.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. -4 and 3 multiply to -12 and add to -1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 4)(x + 3)

4. Solve x^2 - 6x + 8 = 0.

Choices: 2 and 4 · -2 and -4 · 1 and 8 · 3 and 5

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x^2 - 6x + 8 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x - 2)(x - 4).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2 and 4

5. Factor 2x^2 + 7x + 3.

Choices: (2x + 1)(x + 3) · (2x + 3)(x + 1) · (x + 1)(x + 6) · (2x - 1)(x - 3)

Show solution
  1. Challenge: First identify exactly what the question is asking: Factor 2x^2 + 7x + 3.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. The middle terms are 6x and x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (2x + 1)(x + 3)

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