Quadratics and Factoring
A free Algebra I lesson from the “Quadratic Foundations” unit, with a worked example and practice problems including step-by-step solutions.
A quadratic expression has a squared variable term, usually x^2, and its graph is a parabola. Quadratics model situations with curved growth, area, projectile motion, and products of changing quantities. In Algebra I, the main goal is to recognize the structure of a quadratic, factor simple trinomials, and understand that zeros are input values where the expression equals zero. When practicing, look at the first term, middle term, and constant term. A common mistake is treating a quadratic like a linear equation and trying to solve it with only one inverse operation.
What you'll learn
- Recognize quadratic expressions
- Factor simple trinomials
- Use factored form to solve
Worked example
Problem. Factor x^2 + 7x + 10.
- Find two numbers that multiply to 10.
- 5 and 2 multiply to 10 and add to 7.
- Write the factors as (x + 5)(x + 2).
Answer: (x + 5)(x + 2)
Practice problems
1. Which expression is quadratic?
Choices: x^2 + 3x + 2 · 4x + 9 · 7 · x^3 + 1
Show solution
- Warm-up: First identify exactly what the question is asking: Which expression is quadratic?
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- A quadratic has highest power 2.
- x^2 + 3x + 2 is quadratic.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x^2 + 3x + 2
2. Factor x^2 + 5x + 6.
Choices: (x + 2)(x + 3) · (x + 1)(x + 6) · (x - 2)(x - 3) · (x + 5)(x + 1)
Show solution
- Warm-up: First identify exactly what the question is asking: Factor x^2 + 5x + 6.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 2 and 3 multiply to 6.
- 2 and 3 add to 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)
3. For x^2 + 9x + 20, what two positive numbers multiply to 20 and add to 9? Enter the larger one.
Show solution
- Warm-up: 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.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 4 and 5 multiply to 20.
- 4 and 5 add to 9.
- The larger number is 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
4. Factor x^2 + 8x + 15.
Choices: (x + 3)(x + 5) · (x + 1)(x + 15) · (x - 3)(x - 5) · (x + 2)(x + 6)
Show solution
- Core Practice: First identify exactly what the question is asking: Factor x^2 + 8x + 15.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 3 and 5 multiply to 15 and add to 8.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (x + 3)(x + 5)
5. Factor x^2 - 7x + 12.
Choices: (x - 3)(x - 4) · (x + 3)(x + 4) · (x - 2)(x - 6) · (x + 1)(x - 12)
Show solution
- Core Practice: First identify exactly what the question is asking: Factor x^2 - 7x + 12.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 3 and 4 multiply to 12 and add to 7.
- Both signs are negative.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (x - 3)(x - 4)
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