Quadratic equation solver
What is a Quadratic Equation Solver?
A quadratic equation has the form ax² + bx + c = 0 with a not equal to zero. Quadratics appear in physics (projectile motion), finance (break-even analysis), geometry and optimisation problems. This solver finds the roots, real or complex, and shows the discriminant that determines their type.
How to use it
- Enter the coefficients a, b and c.
- Read the discriminant and the roots.
The quadratic formula
x = (-b ± √(b² - 4ac)) / 2a
The discriminant D = b² - 4ac tells you what kind of roots exist:
- D > 0: two distinct real roots; the parabola crosses the x-axis twice.
- D = 0: one repeated real root; the parabola touches the x-axis.
- D < 0: two complex conjugate roots; the parabola does not cross the x-axis.
Worked examples
x² - 5x + 6 = 0: D = 25 - 24 = 1, roots x = 3 and x = 2. Check: (x - 2)(x - 3).x² - 4x + 4 = 0: D = 0, one root x = 2.x² + 2x + 5 = 0: D = -16, roots -1 + 2i and -1 - 2i.
The vertex
The turning point of the parabola is at x = -b / 2a. For x² - 5x + 6 that is x = 2.5, where y = -0.25, the minimum value. This is how you find maximum height, minimum cost or maximum area in applied problems.
Numerical tip
When b² is much larger than 4ac, the standard formula loses precision for one root. Robust solvers compute the larger root first and use x₁ × x₂ = c / a to find the other.
Frequently Asked Questions
What is the discriminant?
The value b squared minus 4ac. It tells you whether the equation has two real roots, one repeated root or two complex roots.
What if a is zero?
Then the equation is linear, not quadratic, and has the single solution x = -c / b if b is not zero.
What are complex roots?
When the discriminant is negative, the roots include the imaginary unit i, the square root of -1, and come as a conjugate pair.
How do I find the vertex of a parabola?
The x-coordinate is -b divided by 2a. Substitute it back into the equation to get the y-coordinate.