Quadratic Formula Calculator

Q: What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a . It finds the values of x that make any quadratic equation (ax² + bx + c = 0) true. You plug in the numbers for a, b, and c, and the formula gives you the solutions.

Q: How do I type an equation in Equation Mode?

Type it like you would write it on paper. Use x^2 for x squared. For example, type 2x^2 - 5x + 3 = 0 . You can also use x² if your keyboard supports it. Make sure to include the = 0 part.

Q: What is the sum and product of roots used for?

The sum of roots equals −b/a and the product of roots equals c/a. These are shortcuts from Vieta's formulas. They let you check your answers quickly or build a quadratic equation when you already know its roots.

Quadratic Formula Calculator

Updated Sep 1, 2026 By Jehan Wadia

Property	Value
Vertex	(1.5, -0.25)
Axis of Symmetry	x = 1.5
Y-intercept	(0, 2)
Sum of Roots	3
Product of Roots	2

Property

Value

Vertex

(1.5, -0.25)

Axis of Symmetry

x = 1.5

Y-intercept

(0, 2)

Sum of Roots

Product of Roots

Introduction

The quadratic formula calculator helps you solve quadratic equations quickly and easily. A quadratic equation is any equation that looks like ax² + bx + c = 0, where a, b, and c are numbers. These equations show up all the time in algebra, and solving them by hand can be tricky. The quadratic formula gives you the answer every time, but plugging in the numbers and doing the math takes work. This tool computes all of that.

Just enter your values for a, b, and c, and the calculator will find the solutions (also called roots) of your equation. It will also tell you the value of the discriminant, which is the part under the square root sign (b² − 4ac). The discriminant tells you how many real solutions your equation has. If it is positive, you get two real solutions. If it is zero, you get one. If it is negative, there are no real solutions, only complex ones. This quadratic formula calculator handles all three cases so you always get a clear answer.

How to Use Our Quadratic Formula Calculator

Enter the three coefficients of your quadratic equation (ax² + bx + c = 0), and this calculator will solve for x using the quadratic formula. It gives you both solutions, including real and complex roots.

Coefficient a: Enter the number in front of x². This is the first term in your equation. The value of "a" cannot be zero, or the equation is not quadratic.

Coefficient b: Enter the number in front of x. This is the middle term in your equation. If there is no x term, enter 0.

Coefficient c: Enter the constant number in your equation. This is the term with no variable. If there is no constant, enter 0.

The quadratic formula is one of the most important tools in algebra. It lets you find the solutions (called roots) of any quadratic equation written in the standard form ax² + bx + c = 0. The formula itself is:

x = (−b ± √(b² − 4ac)) / 2a

A quadratic equation is any equation where the highest power of the variable x is 2. The letters a, b, and c are called coefficients. The value of a controls how wide or narrow the parabola is, b affects where it shifts left or right, and c tells you where the curve crosses the y-axis.

What Is the Discriminant?

The part under the square root sign, b² − 4ac, is called the discriminant. It tells you what kind of answers to expect before you even finish solving:

If the discriminant is positive (greater than 0): The equation has two different real number solutions. The parabola crosses the x-axis at two points.
If the discriminant equals 0: There is exactly one real solution (a repeated root). The parabola just touches the x-axis at one point, called the vertex.
If the discriminant is negative (less than 0): There are no real solutions. Instead, the equation has two complex (imaginary) roots. The parabola never crosses the x-axis at all.

Key Properties of a Quadratic Equation

Every quadratic equation has several useful properties:

Vertex: The highest or lowest point on the parabola. You find the x-coordinate using x = −b / (2a), then plug that back into the equation to get the y-coordinate.
Axis of Symmetry: A vertical line that runs through the vertex at x = −b / (2a). The parabola is a mirror image on both sides of this line.
Y-intercept: The point where the curve crosses the y-axis, which is always at (0, c).
Sum of Roots: Always equals −b / a.
Product of Roots: Always equals c / a.

When Do You Use the Quadratic Formula?

You can solve some quadratic equations by factoring or by completing the square. However, these methods do not always work easily. The quadratic formula works every single time, no matter what the coefficients are. That is what makes it so powerful. Whether the roots are whole numbers, fractions, irrational numbers, or even complex numbers, this formula will always give you the correct answer.

How to Use This Calculator

This calculator gives you two ways to enter your equation. In Coefficient Mode, you type in the values of a, b, and c directly. In Equation Mode, you type the full equation like 2x^2 - 5x + 3 = 0. You can use fractions (like 3/4) or decimals. After you press Calculate, the tool shows you both roots, a complete step-by-step solution, a summary of all key properties, and a graph of the parabola with the roots and vertex marked.

Quadratic equations also appear frequently in physics. For example, projectile motion calculations rely on quadratic equations to determine the trajectory of an object, and free fall problems use them to find how long it takes an object to hit the ground. Displacement also follows a quadratic relationship when acceleration is involved.

Formulas used

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

\Delta = b^2 - 4ac

Vertex

\left( -\frac{b}{2a},\; a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \right)

Axis of Symmetry

x = -\frac{b}{2a}

Sum of Roots

x_1 + x_2 = -\frac{b}{a}

Product of Roots

x_1 \cdot x_2 = \frac{c}{a}

Frequently asked questions

What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It finds the values of x that make any quadratic equation (ax² + bx + c = 0) true. You plug in the numbers for a, b, and c, and the formula gives you the solutions.

What are complex or imaginary roots?

Complex roots happen when the discriminant (b² − 4ac) is negative. You cannot take the square root of a negative number using regular numbers, so the answers include i, which stands for √(−1). Complex roots always come in pairs like 3 + 2i and 3 − 2i.

What does a repeated root mean?

A repeated root means both solutions are the same number. This happens when the discriminant equals zero. On the graph, the parabola just touches the x-axis at one point instead of crossing it at two points.

How do I type an equation in Equation Mode?

Type it like you would write it on paper. Use x^2 for x squared. For example, type 2x^2 - 5x + 3 = 0. You can also use x² if your keyboard supports it. Make sure to include the = 0 part.

What is the sum and product of roots used for?

The sum of roots equals −b/a and the product of roots equals c/a. These are shortcuts from Vieta's formulas. They let you check your answers quickly or build a quadratic equation when you already know its roots.

Can the quadratic formula give irrational answers?

Yes. When the discriminant is positive but not a perfect square, the roots include square roots that cannot be simplified to neat numbers. For example, solving x² − 2 = 0 gives x = √2 and x = −√2, which are irrational numbers. The calculator shows these as decimal approximations.

Quadratic Formula Calculator

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Introduction

How to Use Our Quadratic Formula Calculator