Quadratic Formula Calculator

A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b and c are numbers and a ≠ 0. The quadratic formula is the universal tool for solving it - it works no matter how messy the numbers are. Take x² − 3x − 4 = 0: plug a = 1, b = −3, c = −4 into the formula and you get x = 4 and x = −1 in a few seconds. This quadratic equation solver does that arithmetic for you and, just as importantly, shows every step so you can learn the method, not just copy the answer.

The formula and what each part means

The quadratic formula is one of the most-used results in all of algebra:

Reading it left to right: you negate b, then add and subtract the square root of the discriminant (the piece under the radical, b² − 4ac), and finally divide the whole thing by 2a. The ± sign is the reason a quadratic usually has two answers: one from the plus, one from the minus.

How to use this calculator

You only need the three coefficients. Work through it in order:

1. Put the equation in standard form. Rearrange so one side equals zero: ax² + bx + c = 0. Move every term to the left.
2. Read off a, b and c. a is the number in front of x², b in front of x, and c is the constant. Watch the signs - a minus stays with its term.
3. Type each value into its box. Decimals and negatives are fine. If a term is missing, its coefficient is 0 (for example x² − 9 = 0 has b = 0).
4. Read the results. The roots appear at the top, with the discriminant, vertex and axis of symmetry below, and a step-by-step breakdown you can follow line by line.
5. Check your answer. Substitute a root back into the original equation; it should give zero.

Who this calculator is for

• Algebra 1 & Algebra 2 students learning to solve quadratics and check homework.
• SAT, ACT and GCSE test-takers who want fast, reliable practice and verification.
• College students in calculus, physics or engineering who hit quadratics inside larger problems.
• Teachers and tutors generating worked examples with visible steps.
• Anyone who needs the roots of a quadratic for projectile motion, area, optimization or finance problems.

The discriminant: a shortcut for the number of solutions

Before solving fully, the discriminant D = b² − 4ac tells you what kind of answer to expect:

• D > 0: two distinct real roots - the parabola crosses the x-axis twice.
• D = 0: one repeated real root (a "double root") - the parabola just touches the x-axis at its vertex.
• D < 0: two complex roots (conjugates a ± bi) - the parabola never touches the x-axis.

If D is a perfect square and a, b, c are whole numbers, the roots are rational, meaning the equation could also be solved by factoring.

Discriminant reference table

Use this to interpret the discriminant value at a glance:

Worked example 1: two real roots

Solve x² − 3x − 4 = 0. Here a = 1, b = −3, c = −4.

• Discriminant: D = (−3)² − 4(1)(−4) = 9 + 16 = 25 (positive, so two real roots).
• √25 = 5, so x = (3 ± 5) / 2.
• x = (3 + 5)/2 = 4 and x = (3 − 5)/2 = −1.

Because 25 is a perfect square, the roots are whole numbers and the equation also factors as (x − 4)(x + 1) = 0.

Worked example 2: one repeated root

Solve x² − 6x + 9 = 0. Here a = 1, b = −6, c = 9.

• Discriminant: D = (−6)² − 4(1)(9) = 36 − 36 = 0.
• Since D = 0, x = −b/2a = 6/2 = 3 (a single, repeated root).

This is a perfect-square trinomial: (x − 3)² = 0. The parabola just touches the x-axis at x = 3.

Worked example 3: complex roots

Solve x² + 2x + 5 = 0. Here a = 1, b = 2, c = 5.

• Discriminant: D = (2)² − 4(1)(5) = 4 − 20 = −16 (negative, so complex roots).
• √(−16) = 4i, so x = (−2 ± 4i) / 2.
• x = −1 + 2i and x = −1 − 2i - complex conjugates.

The parabola sits entirely above the x-axis, so there are no real solutions, only complex ones.

The vertex, axis of symmetry and the graph

Every quadratic graphs as a parabola. The vertex is its turning point, located at x = −b/2a; substitute that x back in to get the y-coordinate. The vertical line through the vertex is the axis of symmetry. When a > 0 the parabola opens upward (∪) and the vertex is the minimum; when a < 0 it opens downward (∩) and the vertex is the maximum. The y-intercept is simply (0, c). This calculator reports all four so you can sketch the curve quickly.

Key terms explained

• Coefficient: the numbers a, b and c that multiply the terms.
• Root / solution / zero: a value of x that makes the equation equal zero; where the graph crosses the x-axis.
• Discriminant: b² − 4ac; decides real vs. complex roots.
• Double root: a single solution counted twice, occurring when D = 0.
• Complex conjugate: a pair a + bi and a − bi; quadratics with real coefficients always have complex roots in such pairs.
• Vertex: the maximum or minimum point of the parabola, at x = −b/2a.
• Standard form: the arrangement ax² + bx + c = 0 needed before you read off a, b and c.

Other ways to solve a quadratic

The quadratic formula always works, but it is not the only method:

• Factoring: fastest when the quadratic splits into nice integer factors, e.g. (x − 4)(x + 1) = 0.
• Completing the square: rewrites the equation as (x + p)² = q; it is actually how the quadratic formula is derived.
• Graphing: reading the x-intercepts off a plot - good for estimates, less so for exact answers.

If factoring or completing the square stalls, fall back to the formula - it never fails on a genuine quadratic. When the discriminant is positive but not a perfect square, you can evaluate the leftover square root quickly with the Scientific Calculator.

Where quadratics show up in real life

Quadratics are everywhere once you look. Projectile motion (the height of a thrown ball over time) is a quadratic, and solving it for zero tells you when the object lands. Area and geometry problems - "a rectangle's length is 3 more than its width and its area is 40" - reduce to quadratics. So do optimization questions (maximizing profit or minimizing cost, where the vertex is the answer) and parts of physics, engineering and finance. Knowing the formula means you can finish any of these once you reach the ax² + bx + c = 0 stage.

Tips for getting it right

• Always move everything to one side first so the equation equals zero before identifying a, b and c.
• Keep negative signs attached to their coefficients - a wrong sign on b or c throws off the whole result.
• Compute the discriminant before anything else; its sign tells you immediately what kind of answer to expect.
• Remember the entire numerator −b ± √D is divided by 2a, not just part of it.
• Verify a root by substituting it back into the original equation - the check takes seconds and catches arithmetic slips.

How this compares to related calculators

This page solves quadratic equations specifically. For neighbouring tasks, a sister tool fits better: the Scientific Calculator for general expressions and square roots, the Fraction Calculator when your coefficients are fractions, the Average Calculator for mean/median/mode, the Ratio Calculator for proportions, and the Percentage Calculator for percent problems.

• Coefficients given as fractions? Clear them with the Fraction Calculator before reading off a, b and c.
• Need to evaluate or check a stray square root by hand? The Scientific Calculator handles √, powers and logs.
• Turning a word problem into a percentage or proportion first? Use the Percentage Calculator or the Ratio Calculator, then bring the resulting equation back here.