Quadratic Equation Solver

Q: What does the discriminant tell me?

The discriminant b² − 4ac reveals the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.

Q: What if a is zero?

Then it is not quadratic but linear (bx + c = 0), which has a single solution x = −c ÷ b. The calculator handles that case and tells you.

Q: What is the vertex?

The vertex is the turning point of the parabola, at x = −b ÷ 2a. It is the minimum if a is positive and the maximum if a is negative.

Q: How are complex roots written?

As a real part plus or minus an imaginary part, in the form p ± qi, where i is the square root of −1. They occur when the discriminant is negative.

Solve ax² + bx + c = 0 with the discriminant, roots and vertex.

Roots

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Discriminant

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Nature

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Vertex

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Axis of symmetry

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Enter the coefficients a, b and c. If a is zero the equation is linear, and a single solution is shown.

The quadratic formula

Any quadratic equation can be solved with one formula. The term under the square root — the discriminant — decides whether the solutions are two real numbers, one repeated number, or a complex pair.

x = ( −b ± √(b² − 4ac) ) ÷ 2a

Geometrically, the roots are where the parabola crosses the x-axis. Its turning point, the vertex, sits halfway between the roots on the axis of symmetry, which is why the vertex and roots are closely linked.

Worked example

For x² − 5x + 6 = 0 the discriminant is 1, so there are two real roots: x = 2 and x = 3. The parabola’s vertex sits at (2.5, −0.25).

Reading the results

The discriminant is the quick tell. Positive and you will see two clean crossings of the x-axis; zero and the parabola just touches it; negative and it never reaches the axis, so the roots are complex. The vertex tells you the highest or lowest point either way.

Good to know

Sum and product. The roots add to −b ÷ a and multiply to c ÷ a — a handy check.
Vertex direction. Opens upward when a > 0, downward when a < 0.
Watch the signs. A misplaced minus on b or c is the most common slip.

Frequently asked questions

What does the discriminant tell me?

The discriminant b² − 4ac reveals the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.

What if a is zero?

Then it is not quadratic but linear (bx + c = 0), which has a single solution x = −c ÷ b. The calculator handles that case and tells you.

What is the vertex?

The vertex is the turning point of the parabola, at x = −b ÷ 2a. It is the minimum if a is positive and the maximum if a is negative.

How are complex roots written?

As a real part plus or minus an imaginary part, in the form p ± qi, where i is the square root of −1. They occur when the discriminant is negative.