Quadratic equations are a key part of algebra. To master them, you need more than just theory—you need solid practice. The more problems you solve, the more confident you’ll become.
In this article, you’ll find the top practice problems for quadratics. They cover different solving methods and difficulty levels. Each section includes example problems and the type of technique you should use.
1. Factoring Practice
Factoring is one of the fastest ways to solve quadratics. But it only works when the equation can be factored easily.
Problems
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x² + 7x + 10 = 0
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x² – 9x + 20 = 0
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2x² + 5x + 3 = 0
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x² – 4 = 0
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x² + 2x – 35 = 0
What to Practice
Look for two numbers that multiply to ac and add up to b. Break the middle term if needed and factor by grouping. These problems help build speed and accuracy.
2. Quadratic Formula Practice
Some equations don’t factor neatly. That’s where the quadratic formula comes in. It always works, no matter what the equation looks like.
Problems
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x² + 4x + 1 = 0
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3x² – 2x – 7 = 0
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5x² + x – 6 = 0
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x² – x + ¼ = 0
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2x² + 3x + 9 = 0
What to Practice
Use the formula:
x = (-b ± √(b² – 4ac)) / 2a
Focus on accurate substitution and simplifying square roots. These problems help you get comfortable using the formula and recognizing when to expect real or complex solutions.
3. Completing the Square Practice
Completing the square is useful for solving, graphing, and understanding how quadratic functions behave. It teaches you how to transform equations step by step.
Problems
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x² + 6x + 5 = 0
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x² – 10x + 16 = 0
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2x² + 8x + 6 = 0
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x² + 4x + 1 = 0
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x² – 3x + 2 = 0
What to Practice
Make the coefficient of x² equal to 1, then complete the square by adding and subtracting the correct value. Write the left side as a binomial square and solve by taking square roots. This builds algebraic fluency.
4. Square Root Method Practice
This method is fast but works best when there’s no middle term.
Problems
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x² = 49
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x² = 3
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4x² = 100
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x² – 16 = 0
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(x – 1)² = 9
What to Practice
Isolate the x² term, take the square root of both sides, and don’t forget to include the ± symbol. These are great warm-up problems before tackling more complex ones.
5. Graphing Practice
Graphing helps you understand how quadratics behave visually. These problems focus on roots, direction, and vertex.
Problems
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y = x² – 4x + 3
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y = -x² + 2x + 8
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y = x² + 6x + 5
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y = 2x² – 8x + 6
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y = x² – 1
What to Practice
Plot the graph using a table of values or graphing tool. Identify the roots (x-intercepts), vertex, and axis of symmetry. This builds your understanding of function shape and solution meaning.
How to Use These Problems
Start with factoring problems to warm up. Then move to formulas and completing the square. Use square root problems for speed. End with graphing to see how equations behave.
Try solving five problems a day. Check your answers by substituting the solutions back into the original equations. If you use a calculator or graphing tool, make sure you understand the steps, not just the final result.
Mix up the methods to stay sharp and build flexibility in your thinking.
Final Thoughts
These top practice problems for quadratics cover all major methods—factoring, formulas, square roots, completing the square, and graphing. Each type helps you build a different skill. The more you practice, the more confident you’ll become in solving any quadratic equation you face.
