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Math Practice

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 \(x^{2} - 5x + 6 = 0\).

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    \(a = 1, b = -5, c = 6\); \(D = 25 - 24 = 1\).

    $$ x = \frac{5 \pm 1}{2} \quad\Longrightarrow\quad x = 3 \ \text{ or } \ x = 2 $$

  2. 2.

    Solve \(2x^{2} + 5x - 3 = 0\).

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    \(D = 25 + 24 = 49\).

    $$ x = \frac{-5 \pm 7}{4} \quad\Longrightarrow\quad x = \frac{1}{2} \ \text{ or } \ x = -3 $$

  3. 3.

    Solve \(x^{2} - 4x - 3 = 0\).

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    \(D = 16 + 12 = 28\).

    $$ x = \frac{4 \pm \sqrt{28}}{2} = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7} $$

  4. 4.

    Solve \(3x^{2} - 7x + 2 = 0\).

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    \(D = 49 - 24 = 25\).

    $$ x = \frac{7 \pm 5}{6} \quad\Longrightarrow\quad x = 2 \ \text{ or } \ x = \frac{1}{3} $$

  5. 5.

    Solve \(2x^{2} + x - 6 = 0\).

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    \(D = 1 + 48 = 49\).

    $$ x = \frac{-1 \pm 7}{4} \quad\Longrightarrow\quad x = \frac{3}{2} \ \text{ or } \ x = -2 $$

  6. 6.

    Solve \(x^{2} + 6x + 9 = 0\).

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    \(D = 36 - 36 = 0\).

    $$ x = \frac{-6}{2} = -3 \quad (\text{double root}) $$

  7. 7.

    Solve \(x^{2} = 6x - 4\).

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    Standard form \(x^{2} - 6x + 4 = 0\); \(D = 36 - 16 = 20\).

    $$ x = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5} $$

  8. 8.

    Solve \(4x^{2} + 4x + 1 = 0\).

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    \(D = 16 - 16 = 0\).

    $$ x = \frac{-4}{8} = -\frac{1}{2} \quad (\text{double root}) $$

  9. 9.

    Solve \(3x^{2} + 5x - 2 = 0\).

    Show full solution

    \(D = 25 + 24 = 49\).

    $$ x = \frac{-5 \pm 7}{6} \quad\Longrightarrow\quad x = \frac{1}{3} \ \text{ or } \ x = -2 $$

  10. 10.

    Solve \(6x^{2} - x - 2 = 0\).

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    \(D = 1 + 48 = 49\).

    $$ x = \frac{1 \pm 7}{12} \quad\Longrightarrow\quad x = \frac{2}{3} \ \text{ or } \ x = -\frac{1}{2} $$

  11. 11.

    Solve \(x^{2} + 2x + 5 = 0\).

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    \(D = 4 - 20 = -16\).

    $$ x = \frac{-2 \pm 4i}{2} = -1 \pm 2i $$

  12. 12.

    Solve \(5x^{2} - 2x + 1 = 0\).

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    \(D = 4 - 20 = -16\).

    $$ x = \frac{2 \pm 4i}{10} = \frac{1 \pm 2i}{5} $$

  13. 13.

    Solve \(x^{2} - 2x - 4 = 0\).

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    \(D = 4 + 16 = 20\).

    $$ x = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5} $$

  14. 14.

    Solve \(x^{2} + x - 1 = 0\).

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    \(D = 1 + 4 = 5\).

    $$ x = \frac{-1 \pm \sqrt{5}}{2} $$

  15. 15.

    Without solving, state the number and type of solutions of \(2x^{2} - 3x + 5 = 0\).

    Show full solution

    \(D = 9 - 40 = -31 < 0\): two complex-conjugate solutions.
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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

These problems are hand-written, not auto-generated. For each equation: write standard form, identify , , with signs, compute , substitute, and simplify. Then compare with the solution.

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.

Why is this a curated set rather than auto-generated?+

Most of these answers are irrational (they contain a radical) or complex, which an automatic numeric answer-checker can't verify cleanly. Each problem has a full worked solution instead.

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 Application Practice Problems