ALGEBRA / QUADRATIC

Quadratic Equation Calculator for Roots, Discriminant, Vertex, Steps and Graph

Updated October 13, 2026By Unit Conversion Plus

Find real or complex roots, the discriminant and vertex of a quadratic equation, with worked steps and a graph.

Set up your equation

ax² + bx + c = 0

How to use this solver

Put every term on the left so the right side is zero. Calculate the discriminant D = b² − 4ac, then use x = (−b ± √D) / 2a. Positive D gives two real roots, zero gives a repeated root, and negative D gives two complex roots. The vertex is at x = −b / 2a.

Examples and checks for quadratic equations

Can I find break-even points and a maximum profit?

For the example profit model P(q) = −2q² + 40q − 150, enter −2, 40 and −150 to find where profit is zero. The break-even quantities are 5 and 15. The vertex is (10, 50), so the model’s maximum profit is 50 currency units at a quantity of 10. These conclusions apply only where the assumed model is valid; if q counts items, it must also be a whole number.

Which root should I use in a practical problem?

Check each root against the meaning of x and the original equation. A negative length is unsuitable, but a negative coordinate may be valid. A time before the start of an experiment may be outside the intended interval. Sometimes both roots are useful, as with an object reaching the same height twice.

What do complex roots mean for a measurement problem?

Complex roots mean the equation has no real solution. If x represents a real length, duration or quantity, there is no value that meets the stated conditions under that model. Check your numbers, units and assumptions rather than using just the real part of a complex root.

Should I round the answer before checking it?

Keep the exact fraction or as many digits as possible during the calculation, then round the final measurement to a suitable precision. Substitute the unrounded root into the original equation first. If x counts whole objects, rounding a fractional root may no longer satisfy the equation; check the rounded value separately.

When is a quadratic model useful in everyday calculations?

Quadratics often appear when two dimensions involving the same unknown are multiplied, or when a simplified model contains a squared term. Expand brackets and collect terms before entering a, b and c, with zero on the right. Use consistent units and choose another solver if the highest power is not two.

Check roots without relying only on the graph

For x² − 7x + 10 = 0, the roots 2 and 5 each make the expression zero. Their sum is 7 and product is 10, matching −b/a and c/a. A graph is useful for interpretation, but a small viewing window can hide an intercept and a repeated root may touch the axis without crossing it.

Frequently asked questions

What if the x² coefficient is zero?

The calculator solves the remaining linear equation and explains the reduced degree.

Are the roots exact?

Rational real roots are displayed as exact fractions. Other roots are displayed as decimals, alongside the exact quadratic-formula expression. Complex roots use i, where i² = −1.

Can I type a whole equation?

Use the labelled coefficient fields. First collect like terms and expand brackets yourself. This solver does not parse unrestricted equation text.

How can I find the dimensions of a rectangular garden?

Suppose a garden has an area of 40 m² and its length is 3 m more than its width. Let x be the width. Then x(x + 3) = 40, so enter 1, 3 and −40 for x² + 3x − 40 = 0. The roots are 5 and −8. A width must be positive, so the garden is 5 m wide and 8 m long.

Can I calculate the width of a path around a rectangle?

Suppose a 10 m by 6 m garden has a uniform path around its outside, and the combined area is 96 m². If the path width is x, the outer dimensions are 10 + 2x and 6 + 2x. Expanding (10 + 2x)(6 + 2x) = 96 gives 4x² + 32x − 36 = 0. Enter 4, 32 and −36. The roots are 1 and −9; only a 1 m path is physically meaningful.

Why can a height problem have two positive answers?

An object can pass the same height while rising and while falling. In the simplified model h = −5t² + 20t + 1, height h is in metres and time t is in seconds. To find when h = 16, enter −5, 20 and −15. The roots are t = 1 and t = 3 seconds. Both fit the model: the object passes 16 m on its way up and again on its way down. This example ignores air resistance.