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Quadratic Formula Practice Problems

Fifteen hand-built quadratic equations with full solutions using x = (−b ± √(b² − 4ac)) / 2a. Each solution identifies a, b, c, computes the discriminant, and simplifies the answer.

  1. 1.

    Solve .

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    ; .

  2. 2.

    Solve .

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    .

  3. 3.

    Solve .

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    .

  4. 4.

    Solve .

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    .

  5. 5.

    Solve .

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    .

  6. 6.

    Solve .

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    .

  7. 7.

    Solve .

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    Standard form ; .

  8. 8.

    Solve .

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    .

  9. 9.

    Solve .

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    .

  10. 10.

    Solve .

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    .

  11. 11.

    Solve .

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    .

  12. 12.

    Solve .

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    .

  13. 13.

    Solve .

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    .

  14. 14.

    Solve .

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    .

  15. 15.

    Without solving, state the number and type of solutions of .

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    : two complex-conjugate solutions.
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MPhil in Mathematics, Founder of Mathovia
Published

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Quadratic Formula Practice Problems

The Quadratic Formula lesson covers the formula, the discriminant, and simplifying radical answers; this set is 15 equations to practice on, each fully solved.

How to use this set

For each equation: write standard form, identify , , with signs, compute , substitute, and simplify. Type both roots into the answer box, separated by a comma, and press Check Answers. Exact irrational answers are accepted in the form 2 + sqrt(7), 2 - sqrt(7) or the shorthand 2 ± sqrt(7).

Common mistakes to watch for

Not writing standard form first.

Dropping a negative sign when computing or .

Only dividing part of the numerator by .

Reducing — it does not reduce, because does not divide the .

Where to go next

The formula is completing the square pre-solved — drill that constant with the Completing the Square Practice Problems. Word problems that land here are the Quadratic Applications Practice Problems. For the discriminant table and simplification, see the Quadratic Formula lesson.

Frequently Asked Questions

What is the quadratic formula?+

For , . Both signs give a solution.

What does the discriminant tell me?+

: positive and a perfect square means two rational solutions; positive and not a perfect square means two irrational; zero means one repeated; negative means two complex.

How do I enter an irrational root?+

Exactly, not as a decimal. Type both roots separated by a comma — 2 + sqrt(7), 2 - sqrt(7) — or use the ± shorthand, 2 ± sqrt(7), which the checker expands to the same pair. A repeated root is entered once.

How do I handle the signs of a, b, and c?+

Include the sign with the coefficient. In , so , and .

How do I simplify x = (4 ± √20) / 2?+

Simplify the radical: . Then , dividing every term of the numerator by .

Do I need standard form first?+

Yes. Move every term to one side so the other is zero, then read , , .

More practice problems

Quadratic Equations Practice Problems Completing the Square Practice Problems Quadratic Word Problems Practice