Quadratic Formula Calculator

What the quadratic formula does

A quadratic equation is any equation that can be written as ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and the solutions (also called roots or zeros) are the x-values where the parabola meets the x-axis. The quadratic formula,

x = (−b ± √(b² − 4ac)) ÷ 2a,

gives those solutions for every quadratic, whether or not it factors neatly. It comes from completing the square on the general equation, so it is a shortcut you can always rely on rather than a trick that works only sometimes.

How to use the quadratic formula calculator

1. Rearrange your equation so one side is 0. For example, 3x² = 7x − 2 becomes 3x² − 7x + 2 = 0.
2. Enter a, b and c, including their signs. A missing term has a coefficient of 0, so x² − 9 = 0 has b = 0.
3. The quadratic formula calculator shows the roots immediately. Underneath you’ll find the discriminant, the exact roots in simplest radical form, the vertex, the axis of symmetry, which way the parabola opens, the vertex form and the sum and product of the roots.
4. The working lines show the formula with your numbers substituted, so you can copy the steps into homework or check a hand calculation.

Solving a quadratic by hand

Take 2x² − 3x − 5 = 0, so a = 2, b = −3 and c = −5.

1. Discriminant: b² − 4ac = (−3)² − 4 × 2 × (−5) = 9 + 40 = 49.
2. Square root: √49 = 7.
3. Both roots: x = (3 + 7) ÷ 4 = 2.5 and x = (3 − 7) ÷ 4 = −1.

Check by substituting x = 2.5: 2 × 6.25 − 7.5 − 5 = 0. Because the discriminant was a perfect square, the equation also factors: (2x − 5)(x + 1) = 0.

When the roots are complex

For x² − 6x + 10 = 0, the discriminant is 36 − 40 = −4. There’s no real square root of −4, but √−4 = 2i, where i is the imaginary unit with i² = −1. The roots are x = (6 ± 2i) ÷ 2 = 3 ± i. On a graph, this parabola sits entirely above the x-axis with its lowest point at (3, 1), which is why it has no real zeros.

Simplifying radical answers

When the discriminant isn’t a perfect square, teachers usually expect the exact answer in simplest radical form. For x² − 4x − 1 = 0 the discriminant is 20, and √20 = √(4 × 5) = 2√5. Then x = (4 ± 2√5) ÷ 2 = 2 ± √5, about 4.236068 and −0.236068. The calculator divides out the common factor for you, and the square root calculator shows how any radical simplifies.

What the discriminant tells you

The vertex and the shape of the parabola

Every parabola is symmetric about the vertical line x = −b ÷ 2a. That line passes through the vertex and sits exactly halfway between the two roots. If a is positive the parabola opens upward and the vertex is the minimum value; if a is negative it opens downward and the vertex is a maximum. Rewriting the equation in vertex form, y = a(x − h)² + k, makes the vertex (h, k) visible at a glance. That is why the quadratic formula calculator reports it alongside the roots. The slope of the parabola changes from point to point, unlike a straight line, whose constant slope you can find with the slope calculator.

Real-life uses

• Projectile motion. The height of a thrown object in feet is often modeled as h = −16t² + v₀t + h₀. Setting h = 0 and solving gives the time it lands. The vertex gives the maximum height and when it happens.
• Area problems. A rectangle whose length is 3 m more than its width and whose area is 40 m² gives w(w + 3) = 40, or w² + 3w − 40 = 0, so w = 5 m.
• Break-even and profit. When revenue or cost depends on the square of price or quantity, the roots mark the break-even points and the vertex marks maximum profit.
• Geometry. Problems that combine the Pythagorean theorem with an unknown length often lead to a quadratic equation.

Common mistakes

• Dropping the sign of b. In x² − 7x + 12 = 0, b is −7, so −b is +7. Wrap negative values in parentheses when you substitute.
• Dividing only part of the top by 2a. The whole numerator −b ± √(b² − 4ac) is divided by 2a, not just the square root.
• Forgetting to set the equation to zero. x² + 3x = 10 must become x² + 3x − 10 = 0 before you read off c = −10.
• Squaring a negative b incorrectly. (−5)² is 25, not −25. Typing −5² into a calculator gives −25, because the square is applied before the minus sign.
• Throwing away complex roots. If the problem asks for all solutions, a negative discriminant still gives two answers. They’re complex rather than real.

Glossary

• Root, zero or solution: a value of x that makes ax² + bx + c equal 0.
• Discriminant: b² − 4ac, the expression under the square root.
• Vertex: the turning point of the parabola, at x = −b ÷ 2a.
• Complex conjugates: a pair of numbers p + qi and p − qi.