Quadratic Equation Calculator BR
Solve any quadratic equation ax² + bx + c = 0 — get the real or complex roots, the discriminant and the vertex.
How it works
Enter the coefficients a, b and c. The calculator applies the quadratic formula, reports whether the roots are real or complex, and also gives the discriminant and the vertex of the parabola.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0.
What is the discriminant?
The expression b² − 4ac under the square root. It reveals whether the roots are two real, one repeated, or two complex.
What does a negative discriminant mean?
The equation has no real roots — the two solutions are complex numbers involving the square root of a negative.
Can every quadratic be solved by factoring?
No. Factoring works for neat cases, but the quadratic formula solves every quadratic, including those with irrational or complex roots.
Why must a not equal zero?
If a is zero there is no x² term, so the equation is linear, not quadratic, and the formula does not apply.
What do the two roots represent graphically?
They are where the parabola y = ax² + bx + c crosses the x-axis. One root means it touches the axis; no real roots means it never crosses.
Solving a quadratic equation
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. This calculator finds its roots — the values of x that make it true — using the quadratic formula and shows whether the solutions are real or complex.
The quadratic formula
The roots are given by x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 5x + 6 = 0, that is (5 ± √(25 − 24)) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2.
The discriminant
The part under the square root, b² − 4ac, is the discriminant and tells you the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) roots.
Quadratics describe many real situations — the path of a thrown ball, areas, and profit curves — so being able to solve them quickly is a core algebra skill. Factoring works for neat cases, but the formula always works.
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