Quadratic Equation Calculator

All About Quadratic Equations Calculator

Quadratic equations show up more often than you’d think, from the arc of a basketball shot to the formulas used in physics and business. At first glance, they might just look like more algebra, but learning to solve them means learning to understand curves, changes, and patterns that are everywhere around us. We are here to help you get comfortable with what a quadratic equation really is, how to solve it step by step, and how to use Symbolab’s Quadratic Equation Calculator to make the process clearer, not just faster.

What Is a Quadratic Equation?

Let’s start with the basics—because honestly, that’s where the real understanding begins. A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Here’s what each part means:

1. a is the coefficient of x². It can’t be zero. If it were 0, you wouldn’t have a quadratic at all.

2. b is the coefficient of x (just regular x, not squared).

3. c is the constant term, the number without any x attached.

All three, a, b, and c, are real numbers. And that’s it. That’s the full setup. One variable, squared at most. That’s why we call it a “second-degree” equation.

Real life example:

Imagine you run a small online shop, and you’re trying to figure out how many items to sell to make the most money. Your revenue depends on how many items you sell, but your price drops a little the more you sell, maybe because of discounts or advertising expenses.

You might end up with an equation like:

R(x) = –2x² + 40x

Here, x is the number of items sold, and R(x) is the revenue. This is a quadratic equation. Solving it tells you how many items maximize your revenue, and where the revenue drops to zero (also known as, when you’re not making any money).

1. This equation is in standard form, but quadratics don’t always look like that. You might also see:

2. Factored form: (x – 2)(x – 10) = 0, where the roots (solutions) are clearly visible.

Vertex form: a(x – h)² + k = 0, helpful when graphing or finding the highest or lowest point. Each form has a purpose: standard form is versatile, factored form is straightforward, and vertex form helps you see the graph’s shape. Often, you’ll move between them depending on what the problem needs.

So now that we know what a quadratic equation is, let’s look at what kinds of solutions it can have and what they mean.

How to Solve a Quadratic Equation?

There are four main ways to solve a quadratic equation. Each one works well depending on the situation, and we’ll look at everyday examples that show where you might use them, even outside the classroom.

Factoring

We start with factoring when it’s possible. This method involves rewriting the quadratic expression as a product of two binomials, expressions with two terms (like x – 2 or x + 3). Factoring is quick and neat when the numbers are easy to work with.

Real-life example:

Imagine you’re helping plan a small soccer field for your school project. The total area should be 30 square meters, and you want the width to be 3 meters less than the length.

Let’s call the length x meters. Then the width would be x – 3 meters.

The area of a rectangle is length × width, so we write:

x(x – 3) = 30

x² – 3x – 30 = 0

We solve this equation by factoring:

(x – 6)(x + 5) = 0

This gives two possible solutions: x = 6 or x = –5

Since a length can’t be negative, we use x = 6. That means:

Length = 6 meters

Width = 6 – 3 = 3 meters

So, the field should be 6 meters long and 3 meters wide.

Completing the Square

This method helps when factoring isn’t easy. We turn the equation into a perfect square, which makes it easier to solve.

Real-life example:

You're saving up for a new video game. You keep track of how much you save each week, and the pattern follows this formula:

s(t) = t² + 4t + 1

This tells you how your savings grow over time (t is the number of weeks).

We rewrite it using completing the square:

s(t) = (t + 2)² – 3

This new version helps us find the vertex of the graph. The vertex is the point where something important happens—in this case, where your savings are lowest before they start to grow again. You can think of it as the turning point on a curve: the bottom of a dip or the top of a hill. It helps you understand exactly when your savings will start increasing more quickly.

Using the Quadratic Formula

This formula works for any quadratic equation, even the tricky ones. It’s reliable and always gives us an answer.

The formula is:

x = (–b ± √(b² – 4ac)) / 2a

Real-life example:

In gym class, you throw a basketball into the air. The height of the ball (in meters) after you throw it is given by:

h(t) = –5t² + 10t + 2

You want to know when the ball hits the ground, so you set the height to zero:

–5t² + 10t + 2 = 0

This equation doesn’t factor easily, so we use the quadratic formula to solve it. This tells us exactly when the ball lands. If we graphed it, the ball’s path would make a parabola, a curved shape that rises, reaches a peak, and then falls. The vertex in this case would tell us how high the ball goes, while the solution we just found tells us when it comes back down to the ground.

Graphing

When we graph a quadratic equation, it creates a curved shape called a parabola. The solutions are where the curve touches the x-axis. These are called the x-intercepts.

Real-life example:

You’re planning a school bake sale. You know how many cupcakes you plan to make, and your profit depends on how many you sell. Your profit might follow a pattern like this:

P(x) = –2x² + 12x – 10

If we graph this equation, we can see three important things:

1. The x-intercepts tell us how many cupcakes we need to sell just to break even (where profit is zero).

2. The top of the curve (the vertex) shows us the number of cupcakes that gives the maximum profit. The shape of the graph shows how selling too few or too many affects our profit.

3. Just like a hill, the vertex is the highest point, where profit peaks. After that, adding more cupcakes actually reduces profit, maybe because of extra costs or leftovers.

Common Mistakes While Solving Quadratic Equations and How to Avoid Them

Solving quadratic equations takes practice and sometimes, a small mistake can lead to the wrong answer. Here are some of the most common errors students make and how to avoid them:

Not Setting the Equation to Zero

Mistake: Trying to solve an equation like x² + 5x = 6 without first-moving all terms to one side.

Why it’s a problem: The quadratic formula, factoring, and completing the square only work when the equation is in standard form: ax² + bx + c = 0.

Avoid it: Rewrite the equation as:

x² + 5x – 6 = 0

Mixing Up a, b, and c

Mistake: Plugging the wrong values into the quadratic formula.

Why it’s a problem: One wrong number can completely change the answer.

Avoid it: Identify each coefficient carefully.

For x² – 7x + 10 = 0: a = 1, b = –7, c = 10

Sign Errors

Mistake: Forgetting negative signs or subtracting incorrectly.

Why it’s a problem: Even a small sign error can flip your answer or give a wrong discriminant.

Avoid it: Use parentheses when working with negatives.

(–(–7)) becomes +7

Incorrect Square Root Simplification

Mistake: Simplifying roots like √18 incorrectly (e.g., writing √18 = 3 instead of 3√2).

Why it’s a problem: It leads to wrong or incomplete final answers.

Avoid it: Break down the square root step-by-step. Let Symbolab AI Math Solver guide you by turning on “One step at a time.”

Confusing Complex Numbers

Mistake: Thinking an equation has “no solution” just because the discriminant is negative.

Why it’s a problem: You may miss that the equation actually has two complex (imaginary) solutions.

Avoid it: Look for square roots of negative numbers. Symbolab explains them using i, the symbol for √–1.

Avoiding these mistakes helps you solve more accurately and understand the process better, whether you're working by hand or using a calculator.

Real-Life Applications of Quadratic Equations

Quadratics don’t just live in your math homework. They shape the curves, patterns, and decisions we quietly navigate every day. Here’s where they show up:

Basketball arcs: That perfect shot? It follows a parabola. A quadratic equation can tell you how high the ball goes (the vertex) and when it lands (the x-intercepts).

Bake sale planning: Sell too little, you lose money. Too much, you waste it. Somewhere in between is your sweet spot, maximum profit. A quadratic helps you find it.

Garden design or poster layout: Need to fit a fixed area with changing dimensions? You’ll often end up with a quadratic. Solving it helps you balance length and width.

Physics of falling or flying: Dropped keys, thrown Frisbees, roller coaster drops, it’s all motion that follows a curve. And that curve is often quadratic.

Game design and animation: Characters jumping, arrows flying, or balls bouncing off walls? Behind the scenes, quadratic equations are quietly running the show.

How to Use the Symbolab Quadratic Equation Calculator

The Symbolab Quadratic Equation Calculator helps you solve quadratic equations step by step. Whether you're practicing, checking your homework, or learning how the process works, this tool gives clear explanations at every stage. Here’s how to use it:

Step 1: Enter the Equation

Type your quadratic equation into the input box. You could also upload a screenshot, handwritten equation photo or simply write it in words. If you are a frequent user, you could also install a free Symbolab Chrome extension for convenience.

ax² + bx + c = 0

Example: Let’s use this equation: x² – 7x + 10 = 0. This could be written in words as “x squared minus 7 x plus 10 equals 0”

Step 2: Click “Go”

Once the equation is entered, click the red “Go” button. The Symbolab AI Math Solver will begin working through the solution automatically.

Step 3: View Your Solution and Explore the Options

After you click “Go,” the calculator will instantly display the solution near the center of the screen.

For example, if you entered: x² – 7x + 10 = 0

You’ll immediately see the final answer: x = 5, x = 2

Right below that, you'll find:

1. A “Steps” button to walk through the full solution

2. A “Graph” button to visualize the equation

3. An “Examples” button to try similar problems

On the right side, you’ll also notice chat suggestions like:

1. Explain how to solve quadratic equations

2. Solve by: Completing the square

3. Solve by: Factoring method

These let you choose how you'd like to learn or double-check the method Symbolab used. You can even ask your own question in the chat box below—without needing to sign in or enter personal information.

And if you want to focus on learning one step at a time, just toggle the switch labeled “One step at a time” below the solution. This slows things down so you can really follow each part of the process.

Step 4: View the Step-by-Step Breakdown

Once you click “Go,” Symbolab doesn’t just give you the final answer. It shows a clear, step-by-step breakdown of how the quadratic equation is solved. By default, it may solve the equation using one method (like factoring), but you can view the solution using any of the major solving techniques:

1. Factoring Method
2. Quadratic Formula
3. Completing the Square

Simply use the “Solve by” dropdown menu above the steps to switch between methods. The calculator will instantly update the explanation to show that method in full detail.

All steps are laid out clearly. No skipped work, no missing reasoning. You’ll see how the equation is transformed at each stage, from identifying coefficients to calculating the discriminant, applying formulas, factoring expressions, and reaching the final solutions.

This way, you’re not just seeing what the answer is, you’re understanding how and why it was solved that way, across every method available.

Step 5: View the Graph

Scroll down to the Graph section to see a visual of your quadratic equation.

Symbolab automatically plots the parabola. For the equation:

x² – 7x + 10 = 0

You’ll see a U-shaped curve crossing the x-axis at x = 2 and x = 5—these are the solutions. The lowest point on the curve is called the vertex, showing the minimum value of the function.

Use the zoom tools to adjust your view, or click “View interactive graph” for a closer look. The graph helps you understand not just the solution, but how the equation behaves.

Quadratic equations are fundamental to algebra because they introduce us to non-linear thinking, equations like ax² + bx + c = 0 that curve instead of following a straight line. Solving them through factoring, completing the square, or using the quadratic formula helps us understand key ideas like symmetry, maximum and minimum values, and how graphs relate to equations. These skills aren't just academic, they're essential for exploring functions, modeling real-world situations, and preparing for higher-level math. Symbolab makes the process easier by breaking down each method clearly, step by step. Mastering quadratic equations means mastering the structure and behavior of one of math’s most important patterns.