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Quadratic equation solver: roots with the quadratic formula, step by step

Solve ax² + bx + c = 0 with the quadratic formula: discriminant, real or complex roots, vertex, the steps and the graph of the parabola.

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x² − 5x + 6 = 0

Examples

Roots

x₁ = 2 · x₂ = 3

Δ > 0: two different real roots

Discriminant (Δ)
1
Vertex
(2.5; -0.25)
Sum and product of the roots
5 · 6

Step by step

  1. Δ = b² − 4ac = (-5)² − 4 · 1 · 6 = 1
  2. x = (−b ± √Δ) ÷ 2a = (5 ± 1) ÷ 2
  3. x₁ = 2 · x₂ = 3

The parabola opens upwards when a > 0 and downwards when a < 0; the vertex is its lowest or highest point.

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About this tool

Type the coefficients a, b and c of a quadratic equation and get the roots worked out with the quadratic formula (Bhaskara's formula, as it is known in Brazil). The calculator shows the discriminant Δ = b² − 4ac and what it means - two real roots, one double root or two complex roots -, every step of the calculation, the vertex of the parabola, the sum and product of the roots and a graph with the roots marked. Handy for homework, revision and checking your own working. Everything runs in your browser.

How to use it

  1. 1 Type the coefficients a, b and c (a cannot be zero).
  2. 2 Read the roots and what the discriminant says about them.
  3. 3 Follow the steps of the quadratic formula.
  4. 4 Check the vertex and the graph of the parabola.

Frequently asked questions

What is the discriminant?

Δ = b² − 4ac. If it is positive there are two different real roots; if it is zero, one double root; if it is negative, no real roots (the roots are complex numbers).

What if a = 0?

Then the equation is not quadratic: it is the linear equation bx + c = 0, whose root is x = −c/b.

What are the sum and product of the roots?

For ax² + bx + c = 0 the roots add up to −b/a and multiply to c/a (Viète's formulas) - a quick way to check your answer.

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