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If you took algebra in high school, you probably still have nightmares about using the quadratic formula. It's one of the most complex formulas in high school math. Understanding when and how to use it can seem impossible.

The truth is, the quadratic equation is simpler to solve than it looks. In fact, after using our quadratic formula calculator, you'll wonder why you were ever afraid of it in the first place.

Once you have the equation in standard form, enter the values of a, b, and c into the calculator to generate an instant result. In the guide below, go over some of the most important questions to help simplify one of the most commonly heard of formulas today.


What is a quadratic equation?

Second-order equations (also called quadratic equations) are equations with one variable in which the highest order term is of the second order.

The quadratic formula is used to solve any second-order algebraic equation.


3x2 = 3 is second-order.

0 = 5x2 + 3x + 2 is also second-order.

But 4x3 + 2x2 + x+1 = 0 is not and neither is 2x+1 = 0.

Solving Quadratic Equations

When solving a second-order equation, it's hard to know where to begin. Until you learn about the quadratic formula, that is.

With this formula in your arsenal, you can solve any second-order equation in a matter of seconds.

The Quadratic Formula

The formula looks like this:

x = (-b ± √(b2 - 4ac)) / 2a

If this is the first time you've seen it since high school, it probably looks like gibberish. But don't worry, we're going to walk you through it step by step.

By the time you finish this guide, you'll be an expert.

How do you use the quadratic formula?

As mentioned above, you can use the quadratic formula any time you have a second-order algebraic equation in one variable.

Step One

First, manipulate your equation until you have a zero on one side. You want to put in standard form, which looks like this:

ax2 + bx + c = 0

So for example, if your equation looks like this:

2x2 + 3x - 2 = x+3

Rearrange it so it looks like:

2x2 + 2x - 5 = 0

So, in our case, a = 2, b = 2, and c = -5.

Step Two

Finally, plug these values into the equation like this:

x = (-b ± √(b2 - 4ac))

x = (-2 ± √(22 - 4 * 2 * (-5))
(2 * 2)


x = (-2 ± √(4 - 4 * 2 * (-5))

x = -2 ± √4 - -40

x = -2 ± √44

x = -2 ± 2√11

x = -2 ± 2√11

x = -1 ± √11

This gives you x = -2.158 and x = 1.158. Luckily, you can solve the equation even quicker with a quadratic formula calculator.

How many solutions should I get?

Two solutions

Quadratic equations (usually) have two correct answers. This occurs when b2-4ac, known as the discriminant, is greater than zero.

Since you are taking the square root of the discriminant, any number greater than zero can have two outcomes. The ± sign in the formula gives you the two answers, one for + and one for -.

One solution

Some quadratic equations only have one answer. This occurs when b2-4ac, known as the discriminant, is equal to zero. And therefore,

= (-b ± √0)

= -b

Don't worry if this happens. The square root of zero is zero, which means x has only one possible answer.

No Solutions

It's possible for your equation to have no solutions. This occurs when the discriminant (b2-4ac) is less than zero.

Because you can't take the square root of a negative number, your equation will have no real solutions. It will, however, have "imaginary" solutions". Unless you're in an academic setting, you don't need to worry about those.

When can you skip the quadratic formula?

The quadratic equation is quick and easy to solve, especially if you use our quadratic formula calculator. But sometimes, you can solve your equation even faster using other methods.

C Term = 0

For example, if you have the equation:

x2 + 4x = 0

In other words, your c term is equal to zero and you can see right away that one solution is x = 0. You then divide both sides by x, leaving you with:

x + 4 = 0

So, x = -4 is your other solution.

You could have arrived at the same answer using the quadratic formula, but it may have taken you longer.


Furthermore, if your second-order equation is easy to factor or

a perfect square,
that's often a better way to solve it.

For example, if your equation were:

x2 - 3x + 2 = 0

You could factor it to:

(x-1)(x-2) = 0

And your solutions would be x = 1 and x = 2.

But of course, you need to have experience factoring polynomials. If that's not something you know how to do, you can always use the quadratic equation.

How does that look in a graph?

Consider a graph of f(x). The zeros will be the values of x at which the graph crosses the x-axis.

One Solution

If the quadratic equation only yields one answer, then the function only has one zero. That means the graph touches the x-axis and "bounces" right back.

No Solutions

If the quadratic equation doesn't yield any real answers, then f(x) has no real zeros. That means that the graph never touches the x-axis. It's either entirely above or entirely below it.

Practice, Practice, Practice

Now that you understand the quadratic equation, it's time to use it. After all, the best way to understand a new concept is to apply it over and over again.

The quadratic equation may look intimidating, but once you get the hang of it, you'll realize how simple it is.

So get out there and start using it. You have the power to solve any quadratic equation in the world.

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