Quadratic equation solver
Roots, discriminant, vertex and factored form — with the work shown.
x² + x + = 0
Positive → two real roots
Minimum — parabola opens up
The parabola
x from -0.5 to 3.5. Filled dots are roots; the ring is the vertex.
Working
- Start from the quadratic formula
x = (−b ± √(b² − 4ac)) / 2a - Substitute
x = (−(-3) ± √((-3)² − 4·1·2)) / (2·1) - Evaluate the discriminant
-3² − 4(1)(2) = 1 - Solve
x = 2 or x = 1 - Factored form
(x − 2)(x − 1)
How it works
A quadratic is any equation of the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and its solutions are wherever that parabola crosses the x-axis.
x = (−b ± √(b² − 4ac)) / 2a The discriminant tells you the story before you solve
- b² − 4ac > 0
- Two distinct real roots — the parabola crosses the axis twice.
- b² − 4ac = 0
- One repeated root — the parabola just touches the axis at its vertex.
- b² − 4ac < 0
- No real roots — the parabola misses the axis entirely. The solutions are a complex conjugate pair.
Where the formula comes from
Completing the square. Divide through by a, move the constant across, add (b/2a)² to both sides to make the left a perfect square, then take the square root. The ± appears because a positive number has two square roots — which is the whole reason a quadratic has two answers.
The vertex
The parabola is symmetric about x = −b/2a, so that is where the minimum (a > 0) or maximum (a < 0) sits. This is the form used in optimisation problems: maximum revenue, peak height of a thrown object, least material for a given volume.