Roots Calculator

All About Math Roots Calculator

Some questions in math ask when something reaches its peak. Others ask when it starts or ends. But when you're solving for roots, you're answering a different kind of question: When does this expression equal zero? Roots matter because they mark turning points, boundaries, or breaks. In a graph, they’re where a curve touches or crosses the x-axis. In real life, they can mean when a ball hits the ground, when a budget hits zero, or when two moving objects meet.

In this article, we’ll take our time. You’ll learn what roots really are, how to find them step by step, and how to use Symbolab’s Roots Calculator to check your work or explore more complex equations. No rushing. No shortcuts. Just steady, clear understanding.

What Are Roots in Math?

A root is a value that makes an equation true when you plug it in. More specifically, it’s a number that makes the whole expression equal zero.

Take this equation:

$x^2 - 9 = 0$

The roots are $x = 3$ and $x = -3$, because:

$3^2 - 9 = 0,\quad \text{and} \quad (-3)^2 - 9 = 0$

You’re looking for the values of $x$ that bring the expression back down to zero. That’s what a root does. You might also hear:

• Solution: Another word for root
• Zero of a function: Especially when talking about graphs
• X-intercept: where a graph crosses the x-axis

All three describe the same thing: the input that makes the output zero.

A moment in real life

Picture a basketball tossed into the air. The ball rises, slows, then falls back down. If you model the ball’s height with an equation like:

$h(t) = -16t^2 + 20t + 6$

the roots of that equation tell you when the ball hits the ground. In this case, that’s when $h(t) = 0$. You're solving for $t$, the moment the ball’s height becomes zero. That's a root.

So why zero?

Zero is a turning point. It marks the start, the end, or the pause in a system. It’s where motion stops, profit breaks even, or a signal crosses from positive to negative.

Roots tell us when something reaches that point. And that’s why they matter.

Types of Equations with Roots

Roots can show up in many kinds of equations. Some are straightforward. Others take a little more work. The more you practice, the more you start to notice patterns in the way roots appear.

Let’s look at a few types you’ll most often see.

1. Linear equations

These look like:

$ax + b = 0$

They have one root, because there’s only one $x$ that makes the equation equal zero. You solve it by isolating $x$.

Example:

$2x - 6 = 0 \Rightarrow x = 3$

That’s the root. It’s also the x-intercept if you graphed the line.

Real-life moment:

You’re splitting a restaurant bill evenly. The total is $30$, and you’ve already paid $6$. You want to know how much each of the remaining people should pay:

$2x + 6 = 30$

Solving gives you the fair share.

2. Quadratic equations

These take the form:

$ax^2 + bx + c = 0$

They can have:

• Two real roots
• One real root (if both are the same)
• No real roots (if the graph never touches the x-axis)

You can solve them by factoring, completing the square, or using the quadratic formula.

Example:

x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0 \Rightarrow x = 2,\ 3$

Real-life moment: You're watering a plant, and you want to know when the water level in the pot will drop to zero. The water drains at a changing rate, so the model is curved. The roots tell you when the pot will be empty. 3. Polynomial equations (degree 3 or higher) These equations involve powers of $x$ higher than $2$. Example: $x^3 - x = 0$ You can factor: $x(x - 1)(x + 1) = 0$ So the roots are $x = 0$, $1$, and $-1$. The number of roots depends on the degree (the highest power of $x$). A degree 3 polynomial can have up to 3 real roots. Some roots might be complex. Real-life moment:

You're adjusting the dimmer on your lamp, and it reacts in a nonlinear way. There's a point where the brightness turns off completely. That’s a root. So are other points where the response crosses a neutral threshold.

4. Radical equations

These involve roots, like square roots or cube roots.

Example:

$\sqrt{x} = 4 \Rightarrow x = 16$

Be careful here, check your solution by plugging it back in. Some radical equations can lead to extraneous roots that don’t actually work.

Real-life moment:

You’re rearranging furniture. To find out if a table fits diagonally through a doorway, you use the square root of side lengths. Solving a radical equation tells you whether it’ll make it through or get stuck.

5. Equations with complex roots

If the equation has no real solution (the graph doesn’t cross the x-axis), the roots might be complex.

Example:

$x^2 + 4 = 0$

No real number squared gives $-4$, so:

$x = \pm 2i$

These are still roots, they’re just not real numbers.

Real-life moment:

Not all roots show up on a ruler. You’re trying to model mood swings on a sleep tracker. The data dips below zero, not in the real world, but as part of a curve that cycles over time. The root exists, even if it doesn’t land on a number you can see.

Roots come in many forms. Some sit clearly on the number line. Others live in the complex plane. Either way, they're telling you when the expression equals zero. And that’s always worth paying attention to.

Methods for Finding Roots (Manually)

Finding roots means asking: “What value makes this expression equal zero?” There’s more than one way to get there. The method you use depends on what the equation looks like.

We’ll go through four common approaches. Each one has a different feel, but they all aim for the same thing: finding the point where the output becomes zero.

1. Factoring

This method works best when the expression can be written as a product of simpler expressions.

Example:

$x^2 - 5x + 6 = 0$

We factor:

$(x - 2)(x - 3) = 0$

Then we solve:

$x=2,x=3$

Those are the roots.

Factoring works when the numbers are neat. But not every equation is factorable.

2. Using the Quadratic Formula

When factoring doesn’t work, the quadratic formula always does. For any equation of the form:

$ax^2 + bx + c = 0$

the roots are:

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

Example:

$x^2 + 2x - 3 = 0$

Here, $a = 1$, $b = 2$, $c = -3$

$x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-3)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2}$

$x = \frac{-2 \pm 4}{2} \Rightarrow x = 1,\quad x = -3$

This formula gives exact answers, even when the numbers get messy.

3. Completing the Square

This method rewrites the equation so that one side becomes a perfect square.

Example:

$x^2 + 6x + 5 = 0$

Move the constant:

$x^2 + 6x = -5$

Add $9$ to both sides (half of $6$, squared):

$x^2 + 6x + 9 = 4 \Rightarrow (x + 3)^2 = 4$

Take the square root:

$x + 3 = \pm 2 \Rightarrow x = -1,\quad x = -5$

This method helps when you need to see the structure of the equation more clearly.

4. Graphing

Sometimes, especially when estimating or checking, you can graph the equation and look for where it crosses the x-axis.

Example:

Graph $y = x^2 - 4x - 5$

You’ll see it crosses the x-axis at $x = -1$ and $x = 5$. Those are the roots.

This method doesn’t always give exact answers, but it helps you see the behavior of the equation.

Each of these methods tells the same story in a different way: something hits zero. Something turns. That’s what a root is.

Common Mistakes to Avoid

It’s easy to miss a detail when solving for roots. Most mistakes come from rushing or skipping a step. Here are a few things to keep in mind.

• Not setting the equation equal to zero. You can’t find a root unless the equation is written like $f(x) = 0$.
• Losing a negative sign. Watch signs when factoring or applying the quadratic formula. A small slip can change the whole solution.
• Forgetting both answers when square rooting. If $x^2 = 9$, then $x = 3$ and $x = -3$. Don’t drop one.
• Plugging into the formula wrong. In the quadratic formula, misplacing parentheses or making a small arithmetic error in the square root can throw things off.
• Not checking for extraneous roots. Especially in radical equations, a value might seem to work, but doesn’t hold up when plugged back into the original equation.
• Assuming all equations have real roots. Some don’t. If the discriminant is negative or a square root asks for the square root of a negative number, the roots will be complex.

When something doesn’t feel right, pause. Reread the equation. Run through your steps again. That quiet checking, that’s where the learning actually happens.

How to Use Symbolab’s Roots Calculator

Sometimes you want to be sure you’re right. Other times, you just want to understand how a solution unfolds line by line, with no guesswork. That’s where the Roots Calculator helps.

Here’s how to use it:

1. Start by typing your equation into the box. You can use your keyboard, the math keyboard (good for square roots and exponents), or even upload a photo if your problem’s on paper.
2. Click the red Go button.
3. The calculator finds the roots but more importantly, it shows how they were found. Step by step. You can look at the whole process at once or move through it one piece at a time.
4. And if something feels unclear? Tap Chat with Symbo. You can ask questions about what’s happening in the solution, right when you need help, without leaving the page.

It’s not about shortcuts. It’s about support. Like having someone walk through the steps beside you, not in front of you.

Conclusion

Roots are more than answers. They’re signals that tell you when something changes, when a curve dips below the axis, when a value resets. You’ve seen what they are, how they behave, and how to find them with patience, either by hand or using the Symbolab calculator. So if you’re solving a problem and it feels stuck, don’t worry. Ask: When does this hit zero? Let that guide your next step. Math doesn’t always shout. Sometimes, it taps you on the shoulder. Roots are one way it does that.