Math

Quadratic Equation Solver

Solve Any Quadratic Equation Instantly

Enter your coefficients a, b, and c to solve any quadratic equation of the form ax² + bx + c = 0. This solver uses three methods: the quadratic formula, factoring, and completing the square. Choose the method that works best for your problem, or let the calculator determine the most efficient approach automatically.

Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

This calculator is also known as Quadratic Equation Solver.

Read the complete guide

Which Solving Method Should You Use?

The quadratic formula works universally for any quadratic equation, but factoring is faster when roots are integers, and completing the square is useful for converting to vertex form.

Three Methods Compared

Each method has strengths depending on the equation:

Category	Value
Quadratic Formula	Works for all quadratic equations, including those with complex roots
Factoring	Fastest when roots are integers; limited to factorable equations
Completing the Square	Useful for vertex form conversion and proving the quadratic formula

Examples

Solving a Projectile Motion Problem

A ball is thrown upward from 5 feet with an initial velocity of 48 ft/s. The height equation is h(t) = -16t² + 48t + 5. When does it hit the ground?

Setting h(t) = 0 gives -16t² + 48t + 5 = 0. The discriminant is 48² - 4(-16)(5) = 2624 > 0, so two real roots exist. The positive root t ≈ 3.10 seconds is when the ball hits the ground.

Key takeaway: In projectile problems, the quadratic equation naturally arises from the height equation, and the positive root gives the physically meaningful answer.

Tips for Solving Quadratic Equations

Improve your equation-solving efficiency:

Always rearrange to standard form ax² + bx + c = 0 before identifying coefficients
Check the discriminant first to know how many real solutions to expect
Try factoring first for simple integer coefficients before using the formula
Verify solutions by substituting back into the original equation
For word problems, reject solutions that are physically impossible (negative time, negative length)

Frequently Asked Questions about Quadratic Equation Solver

How do you use the quadratic formula?

The quadratic formula x = (-b ± sqrt(b²-4ac)) / (2a) gives the solutions to any equation ax² + bx + c = 0. Identify a, b, and c from your equation, calculate the discriminant b²-4ac, then substitute into the formula. The ± gives you two solutions.

What if the discriminant is negative?

A negative discriminant means the equation has no real solutions but has two complex conjugate solutions of the form a ± bi. This means the parabola does not cross the x-axis.

Can a quadratic equation have only one solution?

Yes, when the discriminant equals zero, the quadratic has one repeated (double) root. Graphically, this means the parabola touches the x-axis at exactly one point (its vertex).