Quadratic Equation Solver
Solve ax² + bx + c = 0: the roots, the discriminant, the vertex and the factored form.
Good to know. Throw a ball and it traces a parabola. The roots are where it takes off and where it lands.
How it works
In plain words
The quadratic formula gives both solutions at once: x equals minus b, plus or minus the square root of the discriminant, all over 2a. The discriminant, b² − 4ac, says how many real solutions there are: two if it is positive, one if it is zero, none if it is negative.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a vertex = (−b ÷ 2a, c − b² ÷ 4a) ax² + bx + c = a(x − x₁)(x − x₂)
Worked example
For x² − 5x + 6 = 0, the discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2: x = 2 or x = 3. It factors as (x − 2)(x − 3).
Sources
No outside data or rates: this is plain arithmetic.
5 automated tests check its answers on every change.
About the quadratic equation solver
A quadratic equation has the form ax² + bx + c = 0, where a isn’t zero. Its graph is a parabola, and the solutions, or roots, are where the parabola crosses the x-axis. They turn up in physics (a thrown ball), in geometry (a rectangle with a given area and perimeter) and in business (the price that maximises revenue).
Enter a, b and c. For an equation that isn’t in this form yet, move everything to one side first: x² = 5x − 6 becomes x² − 5x + 6 = 0, so a = 1, b = −5 and c = 6. A missing term has the value 0, and x² on its own means a = 1.
The discriminant, b² − 4ac, decides what kind of roots there are. Positive: two different real roots, and the parabola crosses the axis twice. Zero: one repeated root, where the parabola just touches the axis. Negative: no real roots, and the parabola never reaches the axis; the solutions are complex numbers, shown with i, the square root of −1.
The vertex is the parabola’s turning point: its lowest point when a is positive, highest when a is negative. The factored form a(x − x₁)(x − x₂) is shown whenever the roots are real.
Questions people ask
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0.
What does the discriminant tell you?
The number of real solutions: two if b² − 4ac is positive, one if it is zero, and none (two complex ones) if it is negative.
How do I solve x² = 9?
Write it as x² − 9 = 0, so a = 1, b = 0 and c = −9. The solutions are x = −3 and x = 3.