In this post, we’re going to talk about using the quadratic formula to solve quadratic equations. So first, what is the quadratic formula, and when should I use it?

What is the Quadratic Equation?

In short, the quadratic formula is a method for finding the x-intercepts, or the solutions, for a quadratic equation.

When Should I Use It?

You will use the quadratic formula when:

The quadratic equation can’t be factored.
You can’t figure out how to factor the quadratic equation quickly.
The math question requires you to use it.
You are quicker at using the quadratic formula than using your calculator to find the solutions (for tests).

The Formula

The quadratic formula will give you two solutions, which will be the two different x-intercepts for the quadratic equation. Here’s what the quadratic formula looks like:

quadratic formula

Using the Quadratic Formula – Examples

A Simple Example

Now we’re going to try a few examples to get familiar with using this formula. First, let’s try an example that works out nicely. Take a look at this example:

x2 + 4x – 5 = 0

We can actually factor this quadratic equation into (x + 5)(x – 1), which gives us -5 and 1 as the two solutions, but let’s practice using the quadratic formula with this equation.

For the quadratic formula, our “a” will be 1, our “b” will be 4, and our “c” will be -5. Now, let’s plug these numbers into the quadratic formula. When we do that, we’re going to have -4 plus or minus the square root of 4 squared minus 4(1)(-5) divided by 2(1).

Now, let’s solve this equation. First, let’s do the part inside the square root. 4 squared is 16, and 4(1)(-5) equals -20. 16 minus -20 equals 36, and the square root of 36 is 6.

Next, let’s do the denominator. 2(1) equals 2. Now, let’s solve the quadratic equation when we use the plus sign. -4 + 6 = 2, and 2/2 equals 1. So, one of our solutions is 1. Now, let’s solve the equation when we use the minus sign. -4 – 6 = -10, and -10/2 equals -5. So, our second solution is -5. And these are the solutions that we were expecting.

A More Complex Example

Now let’s try one more example, one that doesn’t work out so nicely. Take a look at this example:

2x2 – 5x – 8 = 0

We can’t factor this equation, so let’s use the quadratic formula to solve it. For this equation, our “a” will be 2, our “b” will be -5, and our “c” will be -8. Now, let’s plug these numbers into the quadratic formula. When we do that, we’re going to have 5 plus or minus the square root of -5 minus 4(2)(-8) divided by 2(2).

Now, let’s solve this equation. First, let’s do the part inside the square root. -5 squared is 25, and 4(2)(-8) equals -64. 25 minus -64 equals 89, and the square root of 89 is about 9.43.

Next, let’s do the denominator. 2(2) equals 4. Now, let’s solve the quadratic equation when we use the plus sign. 5 + 9.43 is 14.43, and 14.43/4 equals 3.61. So, one of our solutions is 3.61. Now, let’s solve the equation when we use the minus sign. 5 – 9.43 is -4.43, and -4.43/4 equals -1.11. So, our second solution is -1.11.

So, this is how you use the quadratic formula. Basically, you just need to memorize the formula, know when to use it, and practice using it. It’s important to memorize this formula for tests because most tests won’t give it to you. And finally, with this formula, there is no quadratic equation that you won’t be able to solve.