HisabCalc › Discounts & tips

Quadratic Equation Calculator

Solves the quadratic equation a x^2 + b x + c = 0 with defaults a = 1, b = -3 and c = 2, returning the discriminant, both real roots and the vertex.

Enter your numbers

Coefficient a

Example: 1

Coefficient b

Example: -3

Constant c

Example: 2

Result

Currency

Figures are computed on your device; nothing is sent anywhere.

Formula

D = b^2 - 4ac; roots = (-b +/- sqrt(D)) / (2a); vertex = (-b/2a, c - b^2/4a)

How the calculation works

The calculator reads the three coefficients a, b and c, rejects a value of zero for a, then computes the discriminant D = b^2 - 4ac to decide how many real roots exist.

When the discriminant is zero or positive the two roots come from (-b + sqrt(D)) / (2a) and (-b - sqrt(D)) / (2a), while a negative discriminant stops the run with an error.

The vertex of the parabola sits at x = -b / (2a) and y = c - b^2 / (4a), which for a = 1, b = -3 and c = 2 gives the turning point (1.5, -0.25).

Common mistakes

Forgetting that a cannot be zero is the most frequent error, because the equation stops being quadratic and the division by 2a would be undefined.
A sign slip when subtracting 4ac inside the discriminant flips the result, so check that b^2 - 4ac keeps the original negative value of b intact.
Reporting complex roots for a negative discriminant is wrong here, since this tool is designed to return only the real roots of the equation.

When to use it

Use it to find where a thrown object reaches the ground by solving its height equation, because the real roots mark the times when the height equals zero.
Use the vertex coordinates to locate the maximum or minimum of a profit or cost parabola, which is the turning point of that business model.

Worked example

For a = 1, b = -3 and c = 2, the discriminant is 1, the roots are 2 and 1, and the vertex is at (1.5, -0.25), since x^2 - 3x + 2 factors as (x - 2)(x - 1).

Discriminant 1.00

First root 2.00

Second root 1.00

Vertex x 1.50

Vertex y -0.25

Common questions

Why do I get an error when a is zero?

If a is zero the equation becomes linear instead of quadratic, so the quadratic formula does not apply and the tool refuses to continue.

What does a negative discriminant mean?

A negative discriminant means the parabola never crosses the x-axis, so the equation has no real roots and only complex roots exist.

Last updated: 2026-10-03