Quadratic formula calculator, exact roots, with steps.

The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. This quadratic formula calculator takes a, b and c, or the whole equation typed as you would write it, and gives both roots as exact answers (fractions and simplified square roots, or complex numbers), the discriminant and what it means, the vertex, a factored form when one exists, a graph and the formula worked with your numbers.

How to use the quadratic formula calculator

1. Choose Type a, b and c and fill in the three boxes, or choose Type the equation and write something like 2x^2 + 3x = 5.
2. Read the big cards: the two roots, with decimals. A single card means the root is repeated, and a card with i in it means the roots are complex.
3. The cards below show the discriminant, the vertex, the axis of symmetry, the sum and product of the roots, the y-intercept and the factored form.
4. Open Steps to see the discriminant, the numbers put into the formula, the simplified root and a check. Copy result copies a line with the answer.

What the discriminant tells you

The part under the root, D = b² − 4ac, decides the kind of answer. If D is positive there are two real roots, and the parabola crosses the x axis twice. If D is 0 there is one repeated root, and the parabola just touches the axis. If D is negative the roots are complex numbers and the parabola stays above or below the axis. When D is also a perfect square (1, 4, 9, 49 and so on) the roots are rational and the quadratic factors.

Worked example: rational roots

Solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5, so D = 3² − 4 × 2 × (−5) = 9 + 40 = 49 and √49 = 7. Then x = (−3 ± 7) ÷ 4, which gives x = 4 ÷ 4 = 1 and x = −10 ÷ 4 = −5/2. The factored form is 2(x − 1)(x + 5/2). Check x = 1: 2 + 3 − 5 = 0.

Worked example: roots with a square root

Solve x² + 6x + 7 = 0. D = 36 − 28 = 8 and √8 = 2√2, so x = (−6 ± 2√2) ÷ 2 = −3 ± √2. As decimals that is −1.5858 and −4.4142. The calculator simplifies the root and then cancels the common factor of 2, so you see the shortest exact form.

Worked example: complex roots

Solve x² + 2x + 5 = 0. D = 4 − 20 = −16, and √−16 = 4i, so x = (−2 ± 4i) ÷ 2 = −1 ± 2i. The graph never touches the x axis. The sum of the roots is −2, which equals −b ÷ a, and the product is 5, which equals c ÷ a. Those two rules hold for complex roots too.

Equations that are not in standard form

Type 2x^2 + 3x = 5 and the calculator moves everything to one side: 2x² + 3x − 5 = 0. A missing term just means that coefficient is 0: 3x² − 2 = 0 has b = 0 and gives x = ±√6 ÷ 3. If a is 0 the equation is not quadratic at all, and the page says so.

Common problems

• Sign slips on b. The formula starts with −b. If b = −5 then −b = 5. The steps show the sign you used.
• Dividing only part of the top. The ÷ 2a divides the whole numerator, so (−6 ± 2√2) ÷ 2 becomes −3 ± √2, not −6 ± √2.
• A negative discriminant on a real world problem. It means the situation never happens, for example a ball that never reaches a given height.

To rewrite a quadratic as a(x − h)² + k, use the . To plot the parabola with other functions, use the graphing calculator. To find factors of a whole number, see the factor calculator.