Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Let’s begin by reviewing how to convert between standard and factored form using expressions like the ones we have seen before.

1.

Each row of the table below has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.

Factored form	Standard form
$(x + 5) (x + 6)$
	$x^{2} + 13 x + 30$
$(x - 3) (x - 6)$
	$x^{2} - 11 x + 18$

Compare your answers:

Factored form	Standard form
$(x + 5) (x + 6)$	$x^{2} + 11 x + 30$
$(x + 10) (x + 3)$	$x^{2} + 13 x + 30$
$(x - 3) (x - 6)$	$x^{2} - 9 x + 18$
$(x - 9) (x - 2)$	$x^{2} - 11 x + 18$

In the warm up, you examined factors that had different signs. Two numbers’ signs affect the sum when they are combined and the product when they are multiplied. Use your observations from the warm up to explore these expressions that are in some ways unlike the ones we have seen before.

2.

Each row of the table below has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.

Factored form	Standard form
$(x + 12) (x - 3)$
	$x^{2} - 9 x - 36$
	$x^{2} - 35 x - 36$
	$x^{2} + 35 x - 36$

Compare your answers:

Factored form	Standard form
$(x + 12) (x - 3)$	$x^{2} + 9 x - 36$
$(x - 12) (x + 3)$	$x^{2} - 9 x - 36$
$(x - 36) (x + 1)$	$x^{2} - 35 x - 36$
$(x + 36) (x - 1)$	$x^{2} + 35 x - 36$

3.

Name some ways that the expressions in the second table are different from those in the first table (aside from the fact that the expressions use different numbers).

Compare your answer:

In the second table, the expressions in standard form all have a negative constant term. In the first table, they are all positive and the numbers are smaller.

The factored expressions in the first table are either both sums or both differences (same operation in both binomials). In the second table, they all consist of a sum and a difference (different operations in the binomials).

Why Should I Care?

Illustration of a woman wearing an orange apron arranging a bouquet of flowers at a table, surrounded by vases, flower arrangements, scissors, and string, with a floral picture on the wall behind her.

Matteo's mom is a landscape architect. One day, she took Matteo to see the large succulent garden that she had designed for a new hospital. She told him that the garden's area was determined using a single quadratic function.

Then, she introduced Matteo to Lucy, the accountant for the plant store where she bought the succulents. Lucy used quadratic functions to model a demand curve and determine how to maximize profits. This allowed them to find the most fantastic desert plants that would make the plant store the most money.

Both Matteo's mom and Lucy use quadratic functions to excel at their jobs.

Self Check

Find the factored form of the expression,

x^2− 22x − 48

.

$(x - 8)(x - 16)$
$(x + 2)(x - 24)$
$(x - 2)(x + 24)$
$(x + 12)(x - 4)$

Additional Resources

Interpreting Negative Constant Terms When Factoring Quadratic Expressions

Consider the expression $x^{2} - 34 x + 64$ . (Note the leading coefficient is 1.)

We have factored expressions like this before. We know that when the constant term is positive, the signs of the constants in the factored form must be the same.

Since the middle term is negative, we know that both terms will be negative. So, $x^{2} - 34 x + 64$ factors into $(x - 32) (x - 2)$ .

If the middle term was positive in the expression above, then both constants in the factored form would have been positive.

Example 1

Now consider the expression $x^{2} - 2 x - 24$ . (Note the leading coefficient is 1.)

The constant term is a negative, –24. To find its factored form, we are searching for two numbers that have a negative product. This means that these two numbers must have opposite signs since multiplying a positive number and a negative number yields a negative product.

Now look at the middle term, –2. We know the sum of these numbers must be –2. This means the factor with the larger absolute value must be negative.

Which factors of –24 have a sum of –2? The answer is –6 and 4. Since we know that the larger value must have a negative sign, this confirms the factors we are searching for are −6 and 4.

This gives a factored form of $(x - 6) (x + 4)$ .

Example 2

Find the factored form. (Note the leading coefficient is 1.)

$x^{2} + 7 x - 18$

The constant term is a negative, –18. To find its factored form, we are searching for two numbers that have a negative product.

The middle term is 7. The same two numbers must have a sum of +7.

Which two numbers have a product of –18 and a sum of +7?

The numbers are 9 and –2.

So, the factored form is $(x + 9) (x - 2)$ .

Try it

Try It: Interpreting Negative Constant Terms When Factoring Quadratic Expressions

Find the factored form of each expression.

1. $x^{2} + 7 x - 44$

2. $x^{2} - x - 30$

$(x + 11) (x - 4)$

The factors +11 and –4 yield a product of –44 (the constant term) and sum to be +7 (the coefficient of the linear term).

$(x + 5) (x - 6)$

The factors +5 and –6 yield a product of –30 (the constant term) and sum to be –1 (the coefficient of the linear term).

8.7.2 Interpreting Negative Constant Terms When Factoring Quadratic Expressions

Activity

Additional Resources

Interpreting Negative Constant Terms When Factoring Quadratic Expressions

Try It: Interpreting Negative Constant Terms When Factoring Quadratic Expressions