*Contributor: Michelle Haver. Lesson ID: 13637*

Quadratic functions can model a variety of situations in real life. Let's find out what it means to factor a quadratic expression and what that tells us about the function it defines!

categories

subject

Math

learning style

Visual

personality style

Lion, Beaver

Grade Level

Middle School (6-8), High School (9-12)

Lesson Type

Quick Query

Imagine your teacher took you to the roof of the school, tossed a tennis ball up in the air, and asked you to guess how long it would take for the tennis ball to hit the ground without counting in your head.

- What is your guess?

Back in the classroom, the following equation is written on the board:

h(x) = -16x^{2} + 44x + 42

Your teacher asks you to use the given equation, which gives the height of the ball when *x* seconds have passed, to calculate exactly how long it would take for the ball to hit the ground.

- Can you think of a way to use the equation to answer your teacher's question?

Given that the equation represents the height of the ball when *x* seconds have passed, we want to know how to represent the ball hitting the ground.

- What equation can you set up to represent the ball hitting the ground?

The ground represents a height of 0 feet, and we want to know how many seconds it takes to achieve this. Therefore, we can set the function equal to 0 to try to solve for *x*:

-16x^{2} + 44x + 42 = 0

- How do we solve this equation?

Before we can answer the question above, we need to review the concept of FOILing a factored expression:

To FOIL a factored expression, we multiply the **F**irst, **O**utside, **I**nside, and **L**ast pairs and add them all together. (If you need a review, check out our lesson found under **Additional Resources** in the right-hand sidebar.)

Let's try an example together!

Consider the factored quadratic expression (x+1)(x-2):

Questions:

Factoring a quadratic expression is essentially reversing the operation of FOILing! You can think of this as **unFOILing**!

Let's examine what happens when we FOIL in a little more detail to figure out how exactly to reverse that operation.

In the example above, we ended with the final expression of x^{2} - x - 2.

- Do you notice anything about the relationship between the coefficient of the middle term and the constant term when looking back at the original values from the factored expression (x+1)(x-2)?

Look at the constant values in each of the factors! We have 1 and -2 from each factor. We notice two things:

- The sum of these values gives the coefficient of the middle term of the quadratic.
- The product of these values gives the constant term of the quadratic.

- What does all of this mean?

Imagine that we were first given the expression x^{2} - x - 2 and asked to rewrite it in factored form (in other words, unFOIL the expression).

We would need to figure out what numbers go in the blanks:

(x + _____ ) (x + _____ )

To fill in those blanks, we need to find two numbers which multiply to -2 and add to -1 to get the right coefficients in the original expression!

In other words, we need to find a pair of factors of -2 which sum to -1.

A tip to help you figure this out is to write down all factor pairs of the constant term to see which one adds up to the right value!

In this case, we already know the factored form:

(x + _____ ) (x + _____ )

x^{2} - 1x - 2

Factors of -2: (-1 , 2) and (1, -2)

- Which factors sum to -1?

-1 + 2 = 1 and 1 + (-2) = -1

The factors we need are 1 and -2. Therefore, we can use these to fill in the blanks from before:

(x + 1) (x + - 2)

Let's try a different example together to see if it all makes sense!

Consider the quadratic expression x^{2} + 6x + 8.

To factor this, we want to find two numbers which multiply to 8 and add to 6.

The first step is to list the factors of 8. These factors are the pairs (1,8), (2,4), (-1,-8), and (-2,-4).

Questions:

Now, let's work through factoring the expression!

To factor, we want to fill in the blanks of the following expression with the correct numbers:

(x + _____ ) (x + _____ )

- Based on the first example, what numbers do you think go in the blanks?

If you said, "2 and 4", you are absolutely right!

We get the factored expression (x + 2)(x + 4)! To check our answer, let's FOIL the expression:

(x + 2)(x + 4) = x^{2} + 4x + 2x + 8 = x^{2} + 6x + 8

FOILing the expression gives us the original, so we have successfully factored x^{2} + 6x + 8 as (x + 2)(x + 4).

This is the process you could take to factor quadratic expressions when the leading coefficient is 1.

- What about cases where the leading coefficient is not equal to 1?

To see how to factor quadratic expressions where the leading coefficient is not 1, watch *Learn to Factor Trinomials with Reverse FOIL (3-Minute Math)* from Math Minutes with Mr. White:

Let's try one more example together!

Consider the quadratic expression 10x^{2} - 31x + 15

Questions:

We next want to find the factors of a × c, which are (1,150), (2,75), (3, 50), (6,25), (10,15), (-1,-150), (-2,-75), (-3,-50), (-6,-25), and (-10,-15).

We find that each of these pairs multiplies to 150.

We can now rewrite the expression to split the middle term, which allows us to factor by grouping!

- Can you think of the way to rewrite the expression by filling in the blanks below?

10x^{2} + ____ + ____ + 15

We rewrite the expression as:

10x^{2} + (-6x) + (-25x) + 15

Next, we group the first two and last two terms:

[10x^{2} + (-6x)] + [(-25x) + 15]

Now, you want to factor out the greatest common factor from each of these groups:

2x [5x - 3] + (-5) [5x - 3]

Note that each part now has the same factor (5x - 3), so we can factor that out:

2x | (5x - 3) | - 5 | (5x - 3) | = | (5x - 3)(2x - 5) |

Finally, we see that the factored expression is (5x - 3)(2x - 5). Remember, you can always check your answer by FOILing this expression!

After all of that work, you may be wondering why this even matters! Let's go back to our initial example to explain how this is useful to us.

We are trying to determine how long it takes for the ball to hit the ground when the height is given by the following equation:

h(x) = -16x^{2} + 44x + 42

We previously discussed that the ground represents a height of 0, so we want to solve the equation below for x:

-16x^{2} + 44x + 42 = 0

We can find this information by factoring or unFOILing the quadratic expression!

- Using the information from this lesson, what is the factored form of the quadratic equation?

Once you've factored the equation yourself, check your answer below.

Using the factored form we found, we can rewrite the equation as follows:

(-8x - 6)(2x - 7) = 0

Now, when you multiply two things together and get back 0, at least one of them is equal to 0. This means we have the following:

- | 8x | - | 6 | = | 0 | or | 2x | - | 7 | = | 0 |

We can now solve each equation individually to find the x-values that will make the height of the ball 0:

- | 8x | - | 6 | = | 0 | or | 2x | - | 7 | = | 0 | ||||||||

- | 8x | = | 6 | or | 2x | = | 7 | ||||||||||||

x | = | 6 | or | x | = | 7 | |||||||||||||

8 | 2 |

This gives two possible x-values. These values are called **zeros **or x-intercepts of the function h(x).

- Which of these makes the most sense in the context of what we are trying to find?

If you said x = ^{7}/_{2}, you are correct!

We are trying to find how long it takes the ball to hit the ground, so it makes more sense to choose the positive value. This means it should take 3.5 seconds for the ball to hit the ground!

You've now gone through several examples of how to factor quadratic expressions and how we can use that information to help us answer questions about functions defined with these expressions!

Go on to the *Got It?* section to check your understanding and get some more practice factoring!

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