Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 with exact answers, step-by-step working, discriminant analysis and a parabola graph.
Nudge a coefficient with ▲ ▼ and watch the curve move. Below 0 goes negative; 0 drops a term.
Solving x² − 5x + 6 = 0
x = 2orx = 3
Factored: (x − 2)(x − 3) = 0
Discriminant, vertex and moreΔ = 1 · vertex (2.5, −0.25)
- Discriminant
- Δ = 1
- Nature of roots
- Two distinct real roots
- Vertex (turning point)
- (2.5, −0.25), a minimum
- Axis of symmetry
- x = 2.5
- Vertex form
- y = (x − 2.5)² − 0.25
- Y-intercept
- (0, 6)
- Sum of roots (−b/a)
- 5
- Product of roots (c/a)
- 6
How this works: Method,
How this works
Method
The roots come from the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a: a positive discriminant gives two real roots, zero gives one repeated root and a negative one gives two complex roots. The vertex is at x = −b ÷ 2a, and exact answers (fractions, a factored form and simplified square roots such as √8 = 2√2) are worked out on whole-number coefficients, with decimals of up to 6 places scaled up first.
Changes
- Added a printable report with the parabola, the worked steps and the properties table.
- Redesigned: exact answers as fractions, a factored form and simplified square roots, next to the parabola graph.
- Added a step-by-step solution.
- Results now update as you type, with no Solve button.
- First published.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola, which is a U-shaped curve that opens upward if a > 0 or downward if a < 0.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It provides a direct method to find the solutions (roots) of any quadratic equation. The ± symbol means there are typically two solutions - one using addition and one using subtraction.
What is the discriminant and what does it tell us?
The discriminant is the expression b² - 4ac inside the quadratic formula. It determines the nature of the roots: if discriminant > 0, there are two distinct real roots; if discriminant = 0, there is exactly one real root (a repeated root); if discriminant < 0, there are two complex conjugate roots.
How do you find the vertex of a parabola?
The vertex of a parabola given by y = ax² + bx + c is at the point (h, k) where h = -b/(2a) and k = c - b²/(4a). The vertex represents the maximum point (if a < 0) or minimum point (if a > 0) of the parabola.
Can this solver handle complex (imaginary) roots?
Yes! When the discriminant is negative, the quadratic equation has two complex conjugate roots. This solver calculates and displays them in the form a + bi and a - bi, where i = √(-1). The graph will show that the parabola does not cross the x-axis in this case.
What is vertex form and why is it useful?
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. It's useful because you can immediately identify the vertex coordinates, determine whether the parabola opens up (a > 0) or down (a < 0), and see the maximum or minimum value of the function (k).