What does the discriminant tell me?

It tells you how many real roots the equation has. When it is positive the parabola crosses the x-axis in two places, when it is zero it touches in exactly one, and when it is negative it never crosses — the roots are complex numbers.

Then it is not a quadratic equation — it is bx + c = 0, a straight line with one root at x = −c/b. This calculator requires a ≠ 0.

The turning point of the parabola, where it changes direction from falling to rising (or vice versa). It is the minimum when a is positive and the maximum when a is negative.

Math

Quadratic Formula Calculator

Enter the coefficients a, b and c of ax² + bx + c = 0.

The link carries your figures, so it reopens on exactly these numbers.

The formula

x = (−b ± √(b² − 4ac)) / (2a)

discriminant Δ = b² − 4ac

vertex x = −b / (2a)

vertex y = c − b² / (4a)

Worked example

Solving x² − 5x + 6 = 0

Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
x₁ = (5 + 1) / 2 = 3
x₂ = (5 − 1) / 2 = 2
Vertex at (2.5, −0.25)

Where this goes wrong

Dividing by 2 instead of 2a

The denominator of the quadratic formula is 2a, not 2. When a is not 1, dividing by 2 alone gives the wrong roots. For 2x² − 5x + 3 = 0, dividing by 2 gives x = (5 ± 1)/2, which is 3 and 2 — but the correct roots from dividing by 2(2) = 4 are 1.5 and 1.

Questions

What does the discriminant tell me?: It tells you how many real roots the equation has. When it is positive the parabola crosses the x-axis in two places, when it is zero it touches in exactly one, and when it is negative it never crosses — the roots are complex numbers.
What if a is zero?: Then it is not a quadratic equation — it is bx + c = 0, a straight line with one root at x = −c/b. This calculator requires a ≠ 0.
What is the vertex?: The turning point of the parabola, where it changes direction from falling to rising (or vice versa). It is the minimum when a is positive and the maximum when a is negative.