Quadratic Equation Solver - Roots, Vertex & Steps

A quadratic equation is any equation that can be written in the form ax^2 + bx + c = 0, where a is not zero.

These equations appear throughout algebra, physics, engineering, and economics whenever a relationship involves a squared term. The solutions (called roots) tell you where a parabola crosses the x-axis, when a projectile hits the ground, or where profit reaches zero, among many other applications.

This quadratic formula calculator solves any quadratic equation by applying the quadratic formula. Enter the coefficients a, b, and c, and the calculator returns both roots along with the discriminant value that tells you whether the roots are real and distinct, real and equal, or complex. The step-by-step solution shows every stage of the calculation so you can follow the process or check your manual work.

Whether you are a student learning to solve quadratics, a teacher creating worked examples, or a professional modeling a curved relationship, this tool delivers instant and accurate solutions with complete transparency into the method.

The Quadratic Formula

The quadratic formula provides the solution to any quadratic equation ax^2 + bx + c = 0:

x = (-b +/- sqrt(b^2 – 4ac)) / (2a)

The +/- symbol means there are two solutions: one using addition and one using subtraction. These two values are the roots (or zeros) of the equation.

Example: Solve 2x^2 + 5x – 3 = 0.

a = 2, b = 5, c = -3
Discriminant = b^2 – 4ac = 25 – 4(2)(-3) = 25 + 24 = 49
sqrt(49) = 7
x = (-5 + 7) / 4 = 2/4 = 0.5
x = (-5 – 7) / 4 = -12/4 = -3
Solutions: x = 0.5 and x = -3

Both values can be verified by substituting back into the original equation. At x = 0.5: 2(0.25) + 5(0.5) – 3 = 0.5 + 2.5 – 3 = 0. Correct.

The Discriminant

The discriminant is the expression under the square root in the quadratic formula:

D = b^2 – 4ac

Its value determines the nature of the roots:

D > 0: Two distinct real roots. The parabola crosses the x-axis at two points.
D = 0: One repeated real root (also called a double root). The parabola touches the x-axis at exactly one point (the vertex).
D < 0: No real roots. The parabola does not cross the x-axis. The roots are complex conjugates of the form p + qi and p – qi, where i = sqrt(-1).

This calculator computes and displays the discriminant value and explains its significance for the given equation. When the discriminant is negative, the calculator shows the complex roots.

Other Methods for Solving Quadratics

While the quadratic formula works for every quadratic equation, some equations can be solved more efficiently using other methods.

Factoring. If the equation factors neatly into two binomials, factoring is faster. For example:

x^2 + 5x + 6 = 0 (x + 2)(x + 3) = 0 x = -2 or x = -3

Factoring works well when the roots are integers or simple fractions. For equations with irrational or complex roots, the quadratic formula is necessary.

Completing the square. This method transforms the equation into a perfect square trinomial. It is the algebraic technique used to derive the quadratic formula itself and is important in several advanced applications, including converting equations to vertex form.

x^2 + 6x + 5 = 0 x^2 + 6x = -5 x^2 + 6x + 9 = -5 + 9 (x + 3)^2 = 4 x + 3 = +/-2 x = -1 or x = -5

Graphing. The roots of a quadratic equation are the x-intercepts of the parabola y = ax^2 + bx + c. Graphing provides a visual understanding but is less precise than algebraic methods unless done with software.

The Parabola and Its Properties

Every quadratic equation y = ax^2 + bx + c graphs as a parabola. The key features of the parabola are directly related to the coefficients.

Direction. If a > 0, the parabola opens upward (U-shaped). If a < 0, it opens downward (inverted U).

Vertex. The vertex is the highest or lowest point of the parabola. Its x-coordinate is x = -b/(2a), and its y-coordinate is found by substituting that x value back into the equation. The vertex form of the equation is y = a(x – h)^2 + k, where (h, k) is the vertex.

Axis of symmetry. The vertical line x = -b/(2a) passes through the vertex and divides the parabola into two mirror-image halves.

Y-intercept. The point where the parabola crosses the y-axis is (0, c), found by setting x = 0 in the equation.

X-intercepts (roots). These are the points where y = 0, which is exactly what the quadratic formula solves for.

Applications of Quadratic Equations

Projectile motion. The height of an object thrown or launched follows a quadratic path: h(t) = -16t^2 + v0*t + h0 (in feet, where t is time in seconds, v0 is initial velocity, and h0 is initial height). Setting h(t) = 0 and solving gives the time when the object hits the ground.

Example: A ball is thrown upward at 48 ft/s from a 64-foot height. When does it hit the ground? -16t^2 + 48t + 64 = 0 t^2 – 3t – 4 = 0 (t – 4)(t + 1) = 0 t = 4 seconds (the negative solution t = -1 is not physically meaningful).

Revenue and profit optimization. If revenue is modeled as R(x) = -2x^2 + 100x, where x is price, the maximum revenue occurs at the vertex: x = -100/(2(-2)) = 25. Maximum revenue is R(25) = $1,250.

Geometry. Finding dimensions when area and perimeter relationships lead to quadratic equations. If a rectangle has a perimeter of 28 and an area of 48, the length and width satisfy: l + w = 14 and l x w = 48. Substituting gives l^2 – 14l + 48 = 0, yielding l = 8 or l = 6.

Physics. Kinetic energy, electrical circuits with capacitors, and many other physical systems involve quadratic relationships.

Working with Complex Roots

When the discriminant is negative, the roots involve imaginary numbers. The imaginary unit i is defined as sqrt(-1).

Example: Solve x^2 + 2x + 5 = 0. D = 4 – 20 = -16 x = (-2 +/- sqrt(-16)) / 2 = (-2 +/- 4i) / 2 = -1 +/- 2i

The two roots are -1 + 2i and -1 – 2i. These are complex conjugates. In the real number plane, the corresponding parabola does not cross the x-axis.

Complex roots always come in conjugate pairs for quadratic equations with real coefficients. This calculator shows complex roots when applicable.

Vieta’s Formulas

Vieta’s formulas relate the roots of a quadratic equation to its coefficients without solving it:

For ax^2 + bx + c = 0 with roots r1 and r2:

Sum of roots: r1 + r2 = -b/a
Product of roots: r1 x r2 = c/a

These relationships are useful for checking solutions and for algebraic manipulations where the individual roots are not needed.

Use the Square Root Calculator for discriminant calculations, or the Exponent Calculator for power-related computations.

Frequently Asked Questions

What is the quadratic formula?

x = (-b +/- sqrt(b^2 – 4ac)) / (2a). It solves any equation of the form ax^2 + bx + c = 0. The +/- indicates two solutions: one using addition and one using subtraction.

What is the discriminant?

The discriminant is b^2 – 4ac, the value under the square root in the quadratic formula. If positive, there are two real roots. If zero, there is one repeated root. If negative, the roots are complex (involving imaginary numbers).

Can a quadratic equation have no solution?

Every quadratic equation has two solutions in the complex number system. In the real number system, a negative discriminant means no real solutions. The parabola does not cross the x-axis, and the roots involve imaginary numbers.

How do I factor a quadratic equation?

Find two numbers that multiply to give a x c and add to give b. For x^2 + 5x + 6 = 0, the numbers 2 and 3 multiply to 6 and add to 5, so it factors as (x+2)(x+3) = 0 with roots x = -2 and x = -3. Not all quadratics factor neatly with integers.

What is the vertex of a parabola?

The vertex is the highest or lowest point. Its x-coordinate is -b/(2a). Substitute back into the equation for the y-coordinate. If a > 0, the vertex is the minimum point. If a < 0, it is the maximum.

When do I use the quadratic formula vs factoring?

Use factoring when the equation factors neatly with integer or simple fraction roots. Use the quadratic formula for any equation, especially when roots are irrational or complex. The quadratic formula always works; factoring is a shortcut for simple cases.

What are complex roots?

Complex roots contain the imaginary unit i = sqrt(-1). They occur when the discriminant is negative. For real coefficients, complex roots always come in conjugate pairs: a + bi and a – bi.

Data accurate as of: March 2026