Quadratic equation solver with steps

A quadratic formula calculator that shows the working

This quadratic equation solver handles any equation of the form ax² + bx + c = 0. It works out the discriminant, gives the exact roots as simplified fractions and surds, then their decimal values. It also shows the vertex form, the vertex of the parabola, the factored form, the sum and product of the roots, and draws the graph.

Every line is set out the way a teacher expects to see it in a GCSE or A level answer, or in an Algebra 2 class, so you can check your own working one step at a time.

How to use it

1. Under “Coefficients a, b, c”, type the three numbers: whole numbers, decimals or fractions such as 1/2. An empty box counts as 0.
2. Or pick “Type the equation” and enter it as written, for example 3x^2 = 2x + 1. The tool moves everything to the left and gets 3x² − 2x − 1 = 0.
3. The solutions appear as you type. Read the “Step-by-step solution” below, and tick “Show the sign chart” if you need it.
4. “Copy result” copies everything as plain text. Your last entry is kept on your device.

The discriminant and its three cases

As a discriminant calculator, the tool starts with Δ = b² − 4ac, the part under the square root in the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

• Δ > 0: two different real roots, x₁ = (−b − √Δ) / (2a) and x₂ = (−b + √Δ) / (2a).
• Δ = 0: one repeated root, x = −b / (2a).
• Δ < 0: no real solution. The two roots are complex conjugates, (−b ± i√−Δ) / (2a). For x² + x + 1 = 0, Δ = −3 and the tool shows −1/2 ± (√3/2)i.

A worked example

Solve 3x² − 7x + 2 = 0. Here a = 3, b = −7 and c = 2.

1. Δ = (−7)² − 4 × 3 × 2 = 49 − 24 = 25.
2. Δ > 0 and √25 = 5, so x₁ = (7 − 5) / 6 = 2/6 = 1/3 and x₂ = (7 + 5) / 6 = 2.
3. Check: 3 × 2² − 7 × 2 + 2 = 12 − 14 + 2 = 0.

When Δ is not a perfect square, the answer stays exact. For x² − 3x + 1 = 0, Δ = 5 and the roots are (3 − √5)/2 ≈ 0.382 and (3 + √5)/2 ≈ 2.618. For x² − 12 = 0, Δ = 48, √48 = √(16 × 3) = 4√3, so x = ±2√3.

Vertex form and turning point

Completing the square turns any quadratic into a(x − h)² + k, with h = −b / (2a) and k = −Δ / (4a). The tool writes h as α and k as β. The point (h, k) is the vertex, or turning point. For 2x² − 4x + 1, h = 1 and k = −1, so 2x² − 4x + 1 = 2(x − 1)² − 1 and the minimum value is −1 at x = 1. For the worked example, 3x² − 7x + 2 = 3(x − 7/6)² − 25/12.

Factored form, sum and product

With Δ > 0 the quadratic factors as a(x − x₁)(x − x₂): 3x² − 7x + 2 = 3(x − 1/3)(x − 2), which is the same as (3x − 1)(x − 2). With Δ = 0 you get a(x − x₁)². With Δ < 0 there is no factorisation over the real numbers.

The roots always satisfy x₁ + x₂ = −b/a and x₁ × x₂ = c/a. Here 1/3 + 2 = 7/3 and 1/3 × 2 = 2/3. That is a quick way to check an answer.

What the graph tells you

The solutions are the x-intercepts of the parabola y = ax² + bx + c. The graph marks them with red dots and shows the vertex. Two crossings mean Δ > 0, a single touching point means Δ = 0, and a parabola that never reaches the x-axis means Δ < 0. It opens upwards when a > 0 and downwards when a < 0. The sign chart follows: f(x) has the sign of a outside the roots and the opposite sign between them.

Frequently asked questions

What if a = 0?

Then the equation is linear, bx + c = 0, and the tool solves it anyway: 2x + 4 = 0 gives x = −2. If b is also 0, there is no solution when c ≠ 0, and every number is a solution when c = 0.

Can I enter fractions or decimals?

Yes. 0.5 is read as 1/2 and the working stays exact. For (1/2)x² − x − 4 = 0, Δ = 9 and the roots are −2 and 4.

Why does it show 2√3 instead of 3.4641?

Exam questions usually ask for an exact answer in simplified surd form. The tool takes out the largest square factor, then gives the decimal value to 4 decimal places as well.

How do I type the equation?

Use x^2 or x², and * or nothing for multiplication: 2x^2 - 3x + 1 = 0. Terms can sit on both sides of the = sign, and brackets are accepted.