What Is the Quadratic Formula?

When you have a quadratic equation like ax2 + bx +c = 0, there are several ways to solve for x—one of which is the quadratic formula. What is the quadratic formula? The quadratic formula is:

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what is the quadratic formula - magoosh

As you plug the numbers into this equation and solve for x, here are a few things that you should know to help you.

Quadratic Equation

The basic quadratic equation is ax2+bx+c=0. Some examples are:
2x2 + 7x + 5 = 0 (therefore a = 2, b = 7, c = 5)
x2 + 3x + 2 = 0 (therefore a = 1, b = 3, c = 2)
3x2 + 8x = 0 (therefore a = 3, b = 8, c = 0)
5x + 2x2 − 6 = 0 (therefore a = 2, b = 5, c = -6)

You’ll notice that there won’t always be a number represented for each variable in the quadratic equation. Also, the formula won’t always follow the same order. However, when you notice a similar format, you know that one of the ways to solve for x is through the quadratic formula.

Solving for X

How does one go from the quadratic equation to the quadratic formula? Simply solve for x. As you simplify the equation from ax2 + bx + c = 0, you’ll find that what is the quadratic formula - magooshKnowing the quadratic formula, you just need to input the numbers for a, b, and c and find the answer.

Finding the Answer

There are a few things to remember when solving the equation. First, because of the + and − sign, you’re probably going to end up with two answers. Why is that? Because when graphing the solution, you end up with a curve that typically hits the x-axis twice. So, don’t forget to separate it into two separate equations for the positive and negative signs.

Another thing to remember is the order of operations. As you work through the equations under the square root, remember PEMDAS (parentheses, exponents, multiplication/division, addition/subtraction).

Check out the following example for x2 − 3x − 4 = 0. In this case:
a = 1
b = −3
c = −4

We start by plugging these numbers into the quadratic formula. Then, solve for x:
what is the quadratic formula, a1 - magoosh

what is the quadratic formula, a2 - magoosh

what is the quadratic formula, a3 - magoosh
 
what is the quadratic formula, a4 - magoosh

what is the quadratic formula, a5 - magoosh
 
what is the quadratic formula, a6 - magoosh
x = 4, x = −1

Checking the Answer

The equation can be graphed to determine where the curve hits on the x-axis. Using a graphing calculator, input the equation. You’ll find that the graphing calculator often gives estimates. Therefore, it’s important to use it to check your work rather than to allow it to solve the answer for you. For the above question, your graph would look something like this:
what is the quadratic formula, graph - magoosh
What is the quadratic formula? The quadratic formula is a great tool to help you solve for x. Once you know the formula, just “plug and chug” to determine the answer. Test your knowledge with the following equation and check your answer below:

x2 + 4x − 12 = 0

Answer:
what is the quadratic formula, b1 - magoosh

what is the quadratic formula, b2 - magoosh
 

what is the quadratic formula, b3 - magoosh
 

what is the quadratic formula, b4 - magoosh
 

what is the quadratic formula, b5 - magoosh

x = −6, x = 2

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