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Quadratic Equations: Formula, Use, Examples, and Solutions

Quadratic Equations: Formula, Use, Examples, and Solutions

If you’re just starting to work with quadratic equations, we’re excited for you! That means your algebra adventure is really starting to get interesting (and we do mean “interesting” in a good way!).

That said, we know “interesting” can often start out as “confusing.” If that’s where you find yourself, we’re glad you’re here.

As we start to walk through equations and formulas, it might look overwhelming at first. Allow yourself the time and space to move past that initial shock, and really sit with the information. Trust us: giving yourself a little grace will make a world of difference.

Ready to learn quadratic equations?

What is a quadratic equation?

A quadratic equation is an equation in which the variable is raised to the second power. But what does that really mean? And how do you recognize one on the page?

Maybe you haven’t heard of a variable being “raised to the second power” before, but you’ve heard of a number or variable being “squared” or “raised to the power of $$2$$.” Lucky for you, they all mean the same thing!

A variable raised to the second power will look like this:


Within a quadratic equation, it’ll look like this:

$$3x^2+2x+1=0 $$

That tiny little “$$2$$” is actually hugely important for placing quadratic equations within the greater context of equation types.

Let’s talk about a few:

Second-degree equations

A second-degree equation is a type of equation, and the quadratic equation is considered a second-degree equation. That just means that the greatest power (or exponent) in the equation is $$2$$, like $$x^2$$.

But that’s not all: a quadratic equation is also a polynomial equation! A polynomial equation is also a type of equation. Specifically, it’s an equation made up of variables, coefficients, and exponents.

So, quadratic equations are pretty unique — they’re second-degree polynomial equations. In fact, they’re the only second-degree polynomial equations! Why? Because a quadratic equation is made up of variables, coefficients, and exponents, and the highest exponent is $$2$$.

What is a quadratic in math?

A “quadratic” is also a type of problem; more specifically, it’s one that deals with squaring a variable, or multiplying that variable by itself.

That’s what puts the “quadratic” in “quadratic equation” — because the variable $$x$$ is squared.

Second-order polynomials

A second-order polynomial has all the required elements of a polynomial (variables, coefficients, and exponents) arranged in a very specific format:

$$ax^2 + bx + c = 0$$

The other requirement for a second-order polynomial is that $$a$$ does not equal zero ($$a ≠ 0$$).

Fun fact: The graph of a second-order polynomial is a parabola!

P.S. – Keep an eye on that format. It just might come in handy later.

Now that we know how to identify and classify quadratic equations, let’s get into the quadratic formula. We’ll start with our “why” so that we can keep that in mind as we move forward — and really, isn’t the “why” always the most important part?

What is the quadratic formula used for?

The quadratic formula, as you can imagine, is used to solve quadratic equations.

There are other methods, like factoring or completing the square, but the quadratic formula is usually the most straightforward (and least messy) way to solve a quadratic equation.

And, contrary to popular belief, the quadratic formula does exist outside of math class. In the real world, the quadratic formula can be used for finding the speed of a moving object, studying lenses and curved mirrors, or even charting the path of a rocket launching into space! You might be surprised by how often the quadratic formula is actually used.

Formula for quadratic equations

Okay, so we know why we should embrace the quadratic formula, but how do we use it to solve quadratic equations?

Let’s find out!

Here’s the quadratic formula in all its glory:


The quadratic formula is also sometimes referred to as the “ABC formula,” because we use those $$a$$, $$b$$, and $$c$$ coefficients to help us unlock our solution!

This formula is one of the most efficient ways of solving quadratic equations, so committing it to memory isn’t a bad idea. If you want to learn more about how to use it (with a detailed example!), we can help you over here.

If you’re ready to move on here, let’s take a little bit of a closer look at the quadratic formula:

Looking for “b+-square root b2-4ac 2a”?

As we mentioned, this jumble of Googled letters is called the “ABC formula” because of the coefficients.

First, we identify the coefficients $$a$$, $$b$$, and $$c$$ once the quadratic equation is arranged in standard form. After that, we simply plug those values into the quadratic formula $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$.

Not sure what the standard form of a quadratic equation looks like?

You might surprise yourself!

Standard form of a quadratic equation

As you spend more time with quadratic equations, you’ll notice we talk about standard form a lot — and for good reason. It’s a necessary step of the process!

Remember when we talked about the format of second-order polynomials? That’s actually the standard form of a quadratic equation!

As a refresher, here it is:

$$ax^2 + bx + c = 0$$

Quadratic formula rules

When it comes to working with the quadratic formula and quadratic equations, the main rules you need to keep in mind are actually all the basics from arithmetic operations!

If you’re feeling a little shaky on that foundation, head over here so we can help!

How to find solutions of quadratic equations

Like much of math, there’s more than one way to solve quadratic equations.

We’ve focused on the ABC formula because it’s typically the smoothest and simplest method, but you could also try:

Method Equation format What to do
Factoring $$ax^2+bx+c=0$$ Write the expression as a product of two or more factors
Taking the square root $$x^2=\text{constant}$$ Calculate the square root of both sides of the equation
Completing the square $$ax^2+bx+c=0$$ Add and subtract the same value to/from the expression in order to write it as a perfect square

Did you know you can also just solve for the number of solutions to a quadratic equation? It’s pretty mind-blowing what math can do, isn’t it?

Solve the quadratic equation

We recommend choosing your method from the section below if you want us to walk you through each with more context:

However, if you’re stuck on a problem in front of you, it’s best to scan it with your Photomath app so that we can help YOU with that specific problem — in as much detail as you need.

Getting from quadratic equation to quadratic formula

Staring at a quadratic equation and not sure how to plug it into the quadratic formula?

Remember: you need to write the equation in standard form $$ax^2+bx+c=0$$. Standard form is the bridge between equation and formula, helping you identify which coefficients get plugged into which parts of the formula.

Examples of quadratic equations

You know what time it is: time to practice! It’s hard to truly learn something without actually doing it, so try your hand at these examples:

  1. $$x^2+x-30=0$$
  2. $$5t^2+4t+1=0$$
  3. $$16x^2-4=0$$
  4. $$3x^2+x=0$$
  5. $$5x^2=25$$

Notice yourself getting stuck? Scan the problem with your Photomath app! We can go through each step in all the depth and detail you need, through whichever method you prefer.

Here’s how we solve the first example in the app:

How to derive the quadratic formula

Maybe you’re like us and you’re still curious to know more about the quadratic formula (yes, we do exist).  If that’s you, buckle up! We’re going to walk through how the quadratic formula was derived all those years ago. Here’s the step-by-step:

Step Explanation
$$ax^2+bx+c=0$$ $$\text{This is the initial equation}$$
$$ax^2 + bx = –c$$ $$\text{Subtract the variable } c \text{ from both sides to get rid of the } +c \text{ on the left}$$
$$x^2 + \frac{b}{a}x = –\frac{c}{a}$$ $$\text{Divide both sides by } a \text{ to free } x^2 \text{ of its coefficient}$$
$$x^2 + 2\frac{b}{2a}x = -\frac{c}{a}$$ $$\text{Rewrite } \frac{b}{a} \text{ as } 2\frac{b}{2a}x \text{ so that the second term is } 2pq$$
$$x^2 + 2\frac{b}{2a}x + (\frac{b}{2a})^2= (\frac{b}{2a})^2 -\frac{c}{a}$$ $$\text{Add } (\frac{b}{2a})^2 \text{ on both sides to get a third term of } q^2$$
$$(x + \frac{b}{2a})^2 = (\frac{b}{2a})^2 - \frac{c}{a}$$ $$\text{Use } p^2 + 2pq + q^2 = (p + q)^2 \text{ to simplify the left half of the equation}$$
$$(x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}$$ $$\text{Simplify } (\frac{b}{2a})^2 \text{ on the right and adjust } \frac{c}{a} \text{ to make the denominator } 4a^2$$
$$(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$$ $$\text{Combine the right side into one fraction}$$
$$x + \frac{b}{2a} = \sqrt{\frac{b^2 - 4ac}{4a^2}} \text{ or } x + \frac{b}{2a} = -\sqrt{\frac{b^2 - 4ac}{4a^2}}$$ $$\text{Take the square root on both sides to get two solutions!}$$
$$x = -\frac{b}{2a} + \frac{\sqrt{b^2-4ac}}{2a} \text{ or } x = -\frac{b}{2a} - \frac{\sqrt{b^2-4ac}}{2a}$$ $$\text{Subtract } \frac{b}{2a} \text{ from both sides of the equation}$$
$$x = \frac{-b+\sqrt{b^2-4ac}}{2a} \text{ or } x = \frac{-b-\sqrt{b^2-4ac}}{2a}$$ $$\text{Use } \sqrt{\frac{p}{q}} = \frac{\sqrt{p}}{\sqrt{q}} \text{ on the right side of each equation}$$
$$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$\text{Write the two solutions as one using }\pm \text{ in the numerator. We got the quadratic formula!}$$