Quadratic Formula Calculator with Steps: Exact Answers and Decimals (Years 10 and 10A)

Quadratic formula calculator, with the working out

Enter a, b and c from ax² + bx + c = 0. You’ll get the discriminant and the solutions, in exact form and as decimals, with every step set out the way your teacher wants to see it.

By Ryan, founder of Bayside Academics, 15 years in private tutoring · Updated 6 November 2026

Solve a quadratic equation

Solve $ax^2 + bx + c = 0$

abc

x ≈ 2.58 or x ≈ -0.58

Exact: $x = \dfrac{2 \pm \sqrt{10}}{2}$

Write it as $ax^2 + bx + c = 0$ : $2x^2 - 4x - 3 = 0$ , so $a = 2$ , $b = -4$ , $c = -3$ .
The quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Find the discriminant: $\Delta = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-3) = 16 - (-24) = 40$
$\Delta > 0$ , so there are two solutions.
Substitute: $x = \dfrac{4 \pm \sqrt{40}}{4}$
Simplify the surd: $\sqrt{40} = 2\sqrt{10}$
Divide through by the common factor: $x = \dfrac{2 \pm \sqrt{10}}{2}$
As decimals: $x \approx 2.58$ or $x \approx -0.58$ (2 decimal places)

Rearrange to equal 0 first. If a term is missing, enter 0 (for x² − 9 = 0, b is 0).

The quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Worked example

Solve $2x^2 - 4x - 3 = 0$ . Here $a = 2$ , $b = -4$ , $c = -3$ .

\Delta = (-4)^2 - 4(2)(-3)

= 16 + 24 = 40

x = \frac{4 \pm \sqrt{40}}{4} = \frac{4 \pm 2\sqrt{10}}{4}

= \frac{2 \pm \sqrt{10}}{2}

x \approx 2.58 \text{ or } x \approx -0.58

Simplifying the surd is the step most students miss. See surds explained.

The discriminant: how many solutions?

$\Delta = b^2 - 4ac$	Solutions	Example
Positive	Two	$x^2 + 5x + 6 = 0$ : $\Delta = 1$ , $x = -2$ or $-3$
Zero	One	$x^2 - 6x + 9 = 0$ : $\Delta = 0$ , $x = 3$
Negative	None (real)	$x^2 + 2x + 5 = 0$ : $\Delta = -16$

If $\Delta$ is a perfect square, the quadratic factorises, so factorising may be quicker.

By year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 10

Quadratic equations

Solving quadratics: factorising, the null factor law and the quadratic formula.

Worksheets and topic test · Free PDF with answers

Year 10A

Factorising non-monic quadratics

Quadratics where a isn't 1, ready for senior Methods.

Worksheets and topic test · Free PDF with answers

Quadratic formula: questions

What is the quadratic formula?

For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a.

What does the discriminant tell you?

Δ = b² − 4ac. If it's positive there are two solutions, if it's 0 there's one, and if it's negative there are no real solutions.

When should I use the quadratic formula instead of factorising?

When the quadratic doesn't factorise neatly, or you can't spot the factors quickly. It always works, so it's a safe fallback.

Why does it give a surd answer?

When the discriminant isn't a perfect square, the exact answer contains a surd. Leave it in that form unless the question asks for decimals.

Practise until you don’t need the calculator

Tuterly saves the questions you get wrong, shows the working, and gives fresh questions on the same skill until it’s mastered.

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