Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Shows factoring, completing the square, quadratic formula, discriminant (Δ), vertex, and roots.
Awaiting Input
Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.
Understanding Quadratic Equations
A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. The roots represent the x-intercepts where the parabola crosses or touches the horizontal axis. Depending on the discriminant Δ = b² - 4ac, a quadratic equation may have two distinct real roots, one repeated real root, or two conjugate complex roots.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It provides the exact roots for any quadratic equation in standard form ax² + bx + c = 0.
How do I solve a quadratic equation by factoring?
To factor ax² + bx + c = 0, find two numbers that multiply to ac and add to b. Rewrite the middle term, factor by grouping, and set each factor to zero to solve for x.
What does the discriminant (b² - 4ac) tell you?
If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are two conjugate complex (imaginary) roots.
Can this solver find complex or imaginary roots?
Yes. When the discriminant is negative, the solver automatically calculates complex numbers using the imaginary unit i (where i = √-1) and shows each intermediate step.
How do you solve when a quadratic cannot be factored easily?
When integers do not readily factor, use the quadratic formula or complete the square. Both methods work universally for every quadratic equation.
What is the difference between quadratic and cubic equations?
A quadratic equation has degree 2 (ax² + bx + c = 0) with up to 2 roots. A cubic equation has degree 3 (ax³ + bx² + cx + d = 0) with up to 3 roots, solved using Cardano's formula or synthetic division.
Is this quadratic solver completely free?
Yes, Easy Equate's quadratic equation solver is 100% free with unlimited step-by-step calculations and no registration needed.
Can a quadratic equation have only one solution?
Yes. When the discriminant is exactly zero (b² - 4ac = 0), the quadratic has a single repeated real root (multiplicity 2). Graphically, this occurs when the parabola's vertex touches the x-axis at exactly one point without crossing it (for example, (x - 3)² = 0 gives x = 3).
When should I use completing the square instead of the quadratic formula?
Completing the square is fastest when the leading coefficient is 1 (a = 1) and the linear coefficient (b) is an even number, making it easy to divide by 2 and square. For equations with fractions, odd coefficients, or a ≠ 1, the quadratic formula is generally faster and less prone to arithmetic errors.