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One way to analyze a quadratic expression is to factor it — writing it as the product of two binomials. In this lesson, quadratic expressions with a leading coefficient of will be factored.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Finding a Pair of Numbers

Jordan wants to find a pair of integers with a sum of and a product of
Help Jordan find the pair of numbers.
Discussion

Factoring a Quadratic Trinomial With Leading Coefficient

A quadratic trinomial in the form can be factored as if there exist and such that and

Proof

Suppose that the sum of two numbers and is equal to and their product is equal to
To show substitute the equivalent expressions for and
Factor
Therefore, the quadratic trinomial and the product of the binomials are equal.
As an example, the trinomial below will be factored.
These three steps can be followed to factor it.
1
Analyze the Signs of and
expand_more
For some and the aim is to write the given trinomial as follows.
To do so, the signs of and will be used to determine the signs of and
Here, and so both and are positive.
  • Since is positive, the factors and must have the same sign so that is positive.
  • Since is positive, both and must be positive so that is positive.
As a result, and are positive. To determine the signs in other cases, the following table can be used.
Determining the signs of p and q
2
Find the Pair of Factors of That Has a Sum of
expand_more

It is known that and that and are positive integers. Therefore, two positive factors of whose sum is need to be found. The positive factor pairs of will be listed and the pair with a sum of identified.

Positive Factors of Sum
and
and
and

As seen, the factor pair of and meet these requirements, so the values of and are and

3
Write in Factored Form
expand_more
For the given trinomial, two integers with a sum of and a product of were found.
Therefore, the trinomial can be written as the product of the binomials and
Example

Factoring When and

It's a three-day weekend and Tearrik has the day off from school. He wants to use this time to make a present for his brother's birthday. He bought a nice frame and then chose a photo of the two of them. However, the photo does not fit in the frame, so Tearrik needs to edit it.

A picture of Tearrik and his brother

The editing program represents the area of the photo as and its length as Help Tearrik answer the following questions.

a Write a binomial that represents the width of the photo.
b Tearrik is considering making a new picture frame himself. He needs to know approximately how much material he would need to make the frame. Find the perimeter of the photo if it is inches wide.

Hint

a Start by factoring the given quadratic expression.
b Use the given information to find Then, use the formula for the perimeter of a rectangle.

Solution

a The given quadratic expression represents the area of the photo. Therefore, it should be equal to the product of its length and width.
To find the trinomial must be factored. To do so, identify and and determine the signs of and
Here, and so both and are positive.
  • Since is positive, the factors and must have the same sign so that is positive.
  • Since is positive, both and must be positive so that is positive.

Two positive factors of whose sum is need to be found. Now, list the positive factor pairs of and identify the pair with a sum of

Positive Factors of Sum
and
and
and
and
As seen, the factor pair and satisfies the conditions, meaning that the values of and are and Therefore, the trinomial can be written as the product of the binomials and
Since represents the length of the photo, the must represent the width of the photo.
b The width of the photo is given as inches. Since the binomial represents the width of the photo, the value of can be found.
Before finding the perimeter of the photo, its length must also be computed.
Now, the length and width of the photo are known.
The perimeter can now be calculated by using the perimeter formula.
The photo's perimeter is inches.
Example

Factoring When and

Vincenzo is using the extra day off school to begin building a kennel for his dog. In order to use the garden area in the most effective way, he has to build it on a triangular base. He draws a plan of a right triangle whose hypotenuse is represented by the binomial

Triangle ABC with hypotenuse x+7

It is known that the trinomial is twice the area of the triangle and that the length of is greater than the length of

a What are the possible leg lengths of the triangle when
b Write an expression for the perimeter of the triangle to find the amount of fencing Vincenzo will need.

Hint

a The given trinomial is equal to the product of the leg lengths of the triangle. Factor the given quadratic trinomial.
b The perimeter of a triangle is the sum of all of its side lengths.

Solution

a Since the given trinomial is twice the area of the triangle, the trinomial is equal to the product of leg lengths.
To find and the quadratic trinomial needs to be written as a product of two binomials.
Now identify and to get an idea about the signs of and
For this expression, and so is positive but is negative.
  • Since is negative, the factors and must have opposite signs so that is negative.
  • Since is positive, the factor with a greater absolute value must be positive.

As such, the factor pairs of where one factor is negative should be listed. Then, look for the pair with a sum of

Positive Factors of Sum
and
and
and
and
The factors that satisfy the conditions are and Therefore, the trinomial can be written as the product of the binomials and
These two binomials represent the lengths of the legs. When the expressions will be and
This means that the longer side is and the shorter side is Since is greater than and
b The binomials representing the lengths of the legs are and Additionally, represents the hypotenuse. To find the perimeter in terms of all these binomials should be added.
Example

Factoring When and

Tadeo is using the extra day off school to catch up on homework. He is almost finished with his math homework but is stuck on one problem. He reviews the notes he wrote about factoring the given quadratic trinomial with a leading coefficient of

Tadeo's notes
Describe the error in his notes and help him factor the trinomial correctly so he can spend the rest of the weekend doing something more fun.

Answer

Error: When must be a positive integer and must be a negative integer so that their sum is negative.
Factored Form:

Hint

Start by identifying the values of and for the given quadratic trinomial.

Solution

First, and will be identified.
Here, and meaning that both and are negative.
  • Since is negative, the factors and must have opposite signs so that is negative.
  • Since is negative, the factor with a greater absolute value must be negative so that their sum is negative.

Therefore, Tadeo's first point is correct. If it is assumed that then should be negative. In other words, Tadeo's second point is incorrect.

Tadeo's notes

The given trinomial can be factored using this information. To do so, the factor pairs of where one factor is negative and its absolute value is greater than the other factor are listed. Then, the pair with a sum of should be looked for.

Factors of Sum of Factors
and
and
and
and
and
The factors that satisfy the conditions are and so the trinomial can be written as the product of the binomials and
Example

Factoring When and

An inter-class quiz game is being held at Davontay's school this weekend, and he is his class's champion. The quizmaster Paulina asks Davontay to write a quadratic trinomial and then factor it. The conversation between the quizmaster and Dovantoy is shown in the diagram.

Conversation between quizmaster and Davontay
Help Davontay find the value of and write the trinomial in factored form to win the quiz game.

Hint

If is negative, the sum of two factors of should be negative.

Solution

Davontay needs to write a quadratic trinomial in the form It is also given that is negative and
Since the above trinomial can be factored, the value of should be the sum of the two factors, and , of
Now, two facts can be inferred from this trinomial.
  • Since is positive, the factors and must have the same sign so that is positive.
  • Since is negative, both and should be negative so that is negative.

Based on this information, only negative factor pairs of need to be listed.

Negative Factors of
and
and
and
and
and
and

The sum of the factors in each pair could be the value of so Davontay's concern is valid. There are six values for These values can be found as follows.

Factors of Sum
and
and
and
and
and
and
Of these possible values, is the greatest, which meets Paulina's hint. Finally, the trinomial can be written.
The factors and have a product of and a sum of Therefore, the trinomial can be written as the product of the binomials and
Pop Quiz

Practice Factoring Quadratic Trinomials

Factor each quadratic expression with a leading coefficient Write the answer in such a way that the value of is the greater factor.

Factoring quadratic trinomials
Example

Finding the Dimensions of a Trough by Factoring

Maya and her father spent the long weekend building a trough for the animals on their farm. Her father knows that Maya has a math test coming up soon, so he decides to help her prepare for it by quizzing her about the trough they just built.

Watertrough.jpg

The trough's length is centimeters longer than its width. The area covered by the trough is square centimeters.

a Write an equation for the area of the trough in term of its width and solve the equation by factoring.
b State if the solutions make sense. What are the width and length of the trough?

Answer

a Equation:
Solutions: and
b Only the positive solution makes sense because measures cannot be negative.

Width: centimeters

Length: centimeters

Hint

a Use the formula for the area of a rectangle to write an equation.
b Length cannot be negative.

Solution

a The width of the trough is represented by Since the length of the rectangular trough is centimeters longer than its width, represents the length.
Water trough and its dimensions in terms of w
External credits: Colin Smith
Given that the area of the trough is square centimeters, the following equation can be written using the formula for the area of a rectangle.
Now, the equation needs to be rewritten so that a quadratic trinomial is formed on one side of the equation. Then, it will be solved by factoring.
Rewrite
The quadratic trinomial should be factored.
For this trinomial, and Since the value of is negative, only factor pairs of that have opposite signs will be considered. Since is positive, the factor with a greater absolute value must be positive.
Factors of Sum of Factors
and
and
and
and
and
The factors and satisfy the conditions. Therefore, the trinomial can be written as the product of the binomials and
The left-hand side of the equation is a product of two factors. One of them must be zero so that the product is equal to zero. This is known as the Zero Product Property.
Therefore, the solutions of the equation are and
b Recall the solutions of the equation written in Part A.
Since represents a length, it cannot be negative. This means that only the positive value makes sense. The dimensions of the trough can be found by substituting
Expression
Width
Length

The width of the trough is centimeters and the length is centimeters.

Closure

Finding a Pair of Numbers

The challenge presented at the beginning can now be solved using the information covered in this lesson. Jordan wants to find a pair of integers with a sum of and a product of
Help Jordan find the pair of numbers.

Answer

and

Hint

Isolate one of the variables and substitute it into the other equation. Then, solve the equation by factoring.

Solution

The first step is to isolate in the first equation.
Now, the expression equivalent to is substituted into the other equation. Then, the equation is rewritten so that a quadratic expression is formed on one side of the equation.
Rewrite
To solve this equation, the quadratic trinomial should be factored.
In this trinomial, and Since is positive, factor pairs of that have the same sign should be considered. Of those pairs, negative pairs are listed because is negative.
Factors of Sum of Factors
and
and
and
The factors and has a sum of and a product of The trinomial can now be written as the product of the binomials and
The left-hand side of the equation is a product of two factors. One of them must be zero so that the product is equal to zero. Therefore, is either or by the Zero Product Property.