What does the quadratic formula solve?

The quadratic formula solves equations written as ax^2 + bx + c = 0, where a is not zero.

What does the discriminant mean?

The discriminant b^2 - 4ac tells whether the quadratic has two real roots, one repeated real root, or two complex roots.

Can this calculator handle complex roots?

Yes. If the discriminant is negative, the calculator shows the two complex roots using i notation.

Free algebra calculator

Quadratic Formula Calculator

Solve ax^2 + bx + c = 0 with the quadratic formula, including real roots, repeated roots, complex roots, discriminant, vertex, and axis of symmetry.

Real and complex roots
Discriminant result
Step-by-step work
Vertex and axis

Solution

Roots and discriminant

x^2 - 3x + 2 = 0x = 2

Enter a, b, and c to solve the equation.

Root 1: x = 2
Root 2: x = 1
Discriminant: 1
Root type: Two real roots
Axis of symmetry: x = 1.5
Vertex: (1.5, -0.25)

For homework or tests, show the exact form your teacher expects. This calculator displays practical decimal results and formula steps.

Copy the equation, roots, formula work, and verification checks as plain text.

Formula work

Step-by-step solution

The calculator breaks the answer into the formula setup, discriminant, and final root calculation.

Verification

Check the solution

Root sum: 3 = -b/a
Root product: 2 = c/a

Both checks pass.

How the quadratic formula works

The quadratic formula solves equations in the form ax^2 + bx + c = 0. The coefficients a, b, and c are substituted into x = (-b +/- sqrt(b^2 - 4ac)) / 2a. This is useful when factoring is difficult, when the roots are irrational, or when you want a reliable way to check algebra work.

The value under the square root, b^2 - 4ac, is called the discriminant. It controls the kind of answer the equation has before you even finish solving.

What the discriminant tells you

If the discriminant is positive, the equation has two real roots. If it is zero, the equation has one repeated real root. If it is negative, the equation has two complex roots because the square root of a negative number requires i.

Students often use the discriminant as a quick answer check. For example, if a graph never crosses the x-axis, the discriminant should be negative and the roots should be complex.

Why the vertex and axis are included

The roots show where the parabola crosses the x-axis, but the vertex shows the high or low point of the graph. The axis of symmetry is the vertical line through that vertex. Together, these values make it easier to connect equation solving with graphing.

The axis of symmetry is x = -b / 2a. After finding that x value, the calculator substitutes it back into ax^2 + bx + c to estimate the vertex point.

Common quadratic formula mistakes

The most common mistakes are forgetting parentheses around negative b values, dropping the plus-or-minus sign, and dividing only part of the numerator by 2a. Another common issue is using the formula when a is zero. If a is zero, the equation is linear and should be solved as bx + c = 0.

When you are checking homework, write the substitution step before simplifying. That makes sign mistakes easier to catch.