Free Teacher Resources

Quadratic Solver

Solve ax² + bx + c = 0 with the quadratic formula — every step shown and explained, including the discriminant and simplified radicals.

x² − 5x + 6 = 0
  1. 1

    Identify a, b, and c.

    a = 1, b = -5, c = 6

  2. 2

    Compute the discriminant, b² − 4ac.

    D = (-5)² − 4(1)(6)

    D = 25 − 24 = 1

  3. 3

    Write the quadratic formula and substitute.

    x = ( −b ± √D ) / 2a

    x = ( 5 ± √1 ) / 2

  4. 4

    √D is a whole number, so simplify each root.

    √1 = 1

    x = ( 5 + 1 ) / 2 = 3

    x = ( 5 − 1 ) / 2 = 2

Solution

x = 3 or x = 2

About the Quadratic Solver

The quadratic formula is one of those procedures students can often plug numbers into without ever seeing why each piece matters. This solver takes the coefficients a, b, and c, then walks through the actual sequence a student should follow on paper: identify the values, compute the discriminant, apply the formula, and simplify whatever root type the discriminant points to.

Each step has a toggle that reveals the reasoning behind it, so it can be used either as a quick answer check or as a slowed-down worked example, depending on what the moment calls for.

How to use it in your classroom

  1. Enter the coefficients a, b, and c from a quadratic equation in standard form, ax² + bx + c = 0.
  2. Read the equation echoed back above the steps to confirm it matches what you intended.
  3. Step through the solution: identifying values, computing the discriminant, applying the quadratic formula, and simplifying the result.
  4. Click "Show reasoning" on any step for a plain-language explanation of why that step happens.

Tips from the classroom

  • Set a equal to 1 and pick small integer roots first so students can verify the tool's answer against factoring before trusting it on uglier coefficients.
  • Use a negative discriminant on purpose to show what "no real solutions" looks like in the steps, rather than only ever working examples with clean answers.
  • When the discriminant is a non-perfect square, watch how the tool simplifies the radical — it's worth pausing there since simplifying square roots is often the actual sticking point, not the formula itself.
  • Keep the reasoning toggles closed on a first pass through several problems, then go back and open them only for the step where a student got stuck.

Frequently asked questions

What happens if I enter 0 for a?

The tool won't solve it, since a equation with a equal to zero isn't quadratic. You'll see a prompt asking for a non-zero value of a.

Does it simplify irrational roots, or just give a decimal?

Both — it simplifies the radical (for example, √12 becomes 2√3) and also shows a rounded decimal approximation alongside it.

How does it show that an equation has no real solutions?

When the discriminant works out negative, the steps stop there and the result explicitly states there are no real solutions, rather than trying to force an answer.