What you will be able to do
- Work each problem without looking at the answer.
- Write the algebraic step that justifies each change.
- Check the result in the original relationship.
- Use the error notes to decide what to practise next.
How to use this worksheet
Put each equation in zero form before factoring. Use the zero-product property and verify every root in the original equation.
- Recognise square-root form
- Factor a common factor and trinomials
- Apply the zero-product property
- Connect roots to x-intercepts
For a clean student copy, print this page and cover the worked solutions or select the problem pages in your browser’s print preview.
Practice problems
Problem 1
x² = 49
Problem 2
x² − 5x + 6 = 0
Problem 3
2x² − 8x = 0
Problem 4
x² + 2x − 15 = 0
Problem 5
3x² − 12 = 0
Answer key with worked solutions
Problem 1
x² = 49
- 1.Take both square roots: x = ±√49.
Answer: x = −7 or x = 7
Check: Both squares equal 49.
Problem 2
x² − 5x + 6 = 0
- 1.Factor: (x − 2)(x − 3) = 0.
- 2.Set each factor equal to zero.
Answer: x = 2 or x = 3
Check: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.
Problem 3
2x² − 8x = 0
- 1.Factor the GCF: 2x(x − 4) = 0.
Answer: x = 0 or x = 4
Check: Both make the product zero.
Problem 4
x² + 2x − 15 = 0
- 1.Find numbers with product −15 and sum 2: 5 and −3.
- 2.Factor: (x + 5)(x − 3) = 0.
Answer: x = −5 or x = 3
Check: Expand the factors to recover the original trinomial.
Problem 5
3x² − 12 = 0
- 1.Divide by 3: x² − 4 = 0.
- 2.Factor difference of squares: (x − 2)(x + 2) = 0.
Answer: x = −2 or x = 2
Check: 3(4) − 12 = 0.
Use mistakes to choose the next step
Learn factoring step by stepGraph quadratic functions
| If this happened | Likely issue | Next practice move |
|---|---|---|
| Only positive square root used | ± omitted | When x² = k > 0, consider both signs |
| Zero-product property used before equation equals zero | Structural error | Move all terms to one side first |
| GCF root omitted | Factoring incomplete | Set every factor, including x, equal to zero |
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Useful answers
Frequently asked questions
Why must a quadratic equal zero before factoring?
The zero-product property concludes a factor is zero only when the product equals zero.
Can a quadratic have one real root?
Yes. A repeated root occurs when the graph touches the x-axis at its vertex.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.