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Some of the easiest quadratic expressions to factor are those with a leading coefficient of This lesson will explain how to factor quadratic expressions with leading coefficients other than
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Finding a Pair of Numbers
Consider the numbers and Find two numbers and such that the following conditions are satisfied.
Discussion
Factoring a Quadratic Trinomial
If the greatest common factor (GCF) of a quadratic trinomial of the form is then it can be rewritten by factoring out In this case, since is a factor of both and these two coefficients can be written in terms of
With this information in mind, the expression can be factored.
The leading coefficient of the quadratic expression in the parentheses is Therefore, it can be factored by finding two numbers and such that and
Example
Factoring the Face Area
Magdalena's class is planning a camping trip. Magdalena volunteered to research tents to purchase. When she asked her teacher what size they would buy, he instead told her that the base area of the tent should be square feet and its width is feet.
To choose a suitable tent, Magdalena needs to know the dimensions of the tent's base.
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b What will be the dimensions of the tent when foot?
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Hint
a Think about the greatest common factor of the given quadratic trinomial.
b Substitute the given value into the expressions that represent the width and length of the tent.
Solution
| Factors of | Sum of the Factors |
|---|---|
| and | $\text{-}1+(\text{-}3)=\text{-}4 \ {\color{#FF0000}{\bm{\Large{\times }}}}$ |
| and | $1+3=4 \ {\color{#009600}{\bm{\Large{\checkmark }}}}$ |
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
RearrangeEqn
Rearrange equation
Distr
Distribute
b Now that the equations for the width and length of the tent, substitute into each binomial to find the corresponding values.
| Substitution | Calculation | |
|---|---|---|
| feet | ||
| feet |
The class will buy tents that are feet wide and feet long.
Discussion
Factoring a Quadratic Trinomial
When trying to factor a quadratic trinomial of the form it can be difficult to
Here, and There are six steps to factor this trinomial.
seeits factors. Consider the following expression.
1
Factor Out the GCF of and
expand_more To fully factor a quadratic trinomial, the Greatest Common Factor (GCF) of and has to be factored out first. To identify the GCF of these numbers, their prime factors will be listed.
It can be seen that and share exactly one factor,
Now, can be factored out.
In the remaining steps, the factored coefficient before the parentheses can be ignored. The new considered quadratic trinomial is Therefore, the current values of and are and respectively. If the GCF of the coefficients is this step can be ignored.
2
Find the Factor Pair of Whose Sum Is
expand_more It is known that and so Therefore, the factors must have the same sign. Also, Since the sum of the factors is positive and they must have the same sign, both factors must be positive. All positive factor pairs of can now be listed and their sums checked.
| Factors of | Sum of Factors |
|---|---|
| and | $1+12=13\ {\color{#009600}{\bm{\Large{\checkmark }}}}$ |
| and | $2+6=8\ {\color{#FF0000}{\bm{\Large{\times }}}}$ |
| and | $3+4=7\ {\color{#FF0000}{\bm{\Large{\times }}}}$ |
In this case, the correct factor pair is and The following table sums up how to determine the signs of the factors based on the values of and
| Factors | ||
|---|---|---|
| Positive | Positive | Both positive |
| Positive | Negative | Both negative |
| Negative | Positive | One positive and one negative. The absolute value of the positive factor is greater. |
| Negative | Negative | One positive and one negative. The absolute value of the negative factor is greater. |
Such analysis makes the list of possible factor pairs shorter.
3
Write as a Sum
expand_more The factor pair obtained in the previous step will be used to rewrite the term — the linear term — of the quadratic trinomial as a sum. Remember that the factors are and
The linear term can be rewritten in the original expression as
4
Factor Out the GCF of the First Two Terms
expand_more The expression has four terms, which can be grouped into the and the Then, the GCF of each group can be factored out.
The first two terms, and can be factored.
The GCF of and is
5
Factor Out the GCF of the Last Two Terms
expand_more 6
Factor Out the Common Factor
expand_more If all the previous steps have been performed correctly, there should now be two terms with a common factor.
The common factor will be factored out.
The factored form of is Remember that the original trinomial was and that the GCF was factored out in Step This GCF has to be included in the final result.
Example
Factoring When is Positive
Heichi and Maya are reading a pamphlet about local campsites. They volunteered to find a suitable campground for their class field trip. Their math teacher, who goes camping often, gives them a challenge to help them decide where to go. If they answer his questions about the pamphlet correctly, he will tell them about his favorite local campsite.
Their teacher tells them that the area of the pamphlet is square inches and that it has a width of inches.
a Write a binomial that represents the length of the pamphlet.
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b Find the perimeter of the pamphlet if the width is inches.
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Solution
- The Greatest Common Factor (GCF) of and is
- Since which is greater than zero, the factors must have the same sign.
- Since which is negative, the sum of the factors will be negative. Therefore, both of the factors must be negative.
Now, all the negative factor pairs of will be listed and checked whether their sum is
| Factors of | Sum of Factors |
|---|---|
| and | |
| and |
CommutativePropAdd
Commutative Property of Addition
FactorOut
Factor out
Rewrite
Rewrite as
FactorOut
Factor out
FactorOut
Factor out
b To find the perimeter of a rectangle, all four side lengths must be added. The width of the pamphlet is given as inches. Since the binomial represents the width, the value of can be found from here.
Example
Factoring When is Negative
At the campsite, each camper is assigned a rectangular area of square feet for their tent. Dominika wants to place a tarp on the ground below her tent to make sure it stays dry underneath. She does not remember the size of the tarp she has, but she knows that its length is inch less than twice its width. 
Dominika finds that the tarp fits exactly in the allocated area. Help her find the length and width of the tarp. 
Since the base is a rectangle, its area is equal to the product of its width and length.
Now, the obtained quadratic expression will be factored. First, its coefficients will be identified.
Here, and With this in mind, the following conclusions can be made.
Of these factors, and satisfy the conditions. Therefore, the term of the trinomial on the left-hand side of the equation will be rewritten by using these numbers as a sum.
Then, the obtained expression will be factored by grouping.
Since the product of the two binomials is zero, at least one binomial must equal zero. With this in mind, the Zero Product Property will be applied.
The solutions to the equation are and Since a negative length does not make sense, cannot be negative. This means that the value of the width is feet. With this information and knowing that the length is foot less than twice the width, the length can be found.
The base of the tarp has a length and a width of and feet, respectively.
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Hint
Start by writing a quadratic expression for the area of the tarp. Then, factor the obtained expression.
Solution
An expression for the area of the tarp can be written by using the expressions for its length and width. Let be the width of the tarp. Since the length is inch less than twice the width, it can be expressed as
\(A=w \ell\)
SubstituteII
$A={\color{#0000FF}{28}}$, $\ell={\color{#009600}{2w-1}}$
\({\color{#0000FF}{28}}=w ({\color{#009600}{2w-1}})\)
▼
Rewrite
\(2w^2-w-28=0\)
- The Greatest Common Factor (GCF) of and is
- Since which is less than zero, the factors must have the different signs. Therefore, there is a positive factor and a negative factor.
- Since the sum of the factors is negative. This means that the absolute value of the negative factor is greater than the absolute value of the positive factor.
Now, the factor pairs of with opposite signs and where the absolute value of the negative factor is greater than the absolute value of the positive number will be listed. Also, their sum will be calculated to see if it is
| Factors of | Sum of Factors |
|---|---|
| and | |
| and | |
| and | |
| and |
\(2w^2-8w+7w-28=0\)
FactorOut
Factor out
SplitIntoFactors
Split into factors
FactorOut
Factor out
FactorOut
Factor out
▼
Solve using the Zero Product Property
Example
Factoring When is Negative
Dylan is practicing on the basketball court at the campground. For the ball to fall through the hoop without hitting the rim or backboard, it needs to follow a specific path after being thrown at a angle.
Error loading image.
This path can be modeled with a quadratic function.
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Hint
The value of is zero when the ball hits the ground.
Solution
The height of the ball is zero when the it hits the ground. Therefore, to find the total time it takes for the ball to hit the ground, will be substituted for in the given equation.
Now, the trinomial will be factored. First, its coefficients will be identified.
Here, and With this information in mind, the following conclusions can be made.
The factors of whose sum is are and The term of the right-hand side can be rewritten as a subtraction of these numbers.
Now, the obtained expression will be factored by grouping.
The factoring process is finally done. Now, the Zero Product Property can be used to find the values.
Since a negative time does not make sense, is the only solution that satisfies the conditions. This means that the ball will hit the ground seconds after it is thrown.
- The Greatest Common Factor (GCF) of and is
- Since which is less than zero, the factors must have different signs. Therefore, one of the factors is positive and one of the factors is negative.
- Since the sum of the factors is positive. Therefore, the absolute value of the positive factor is greater than the absolute value of the negative factor.
Now, the factor pairs of with opposite signs and where the absolute value of the positive factor is greater than the absolute value of the negative factor will be listed. Also, their sum will be calculated to see whether it is
| Factors of | Sum of Factors |
|---|---|
| and | |
| and | |
| and | |
| and |
RearrangeEqn
Rearrange equation
FactorOut
Factor out
Rewrite
Rewrite as
FactorOut
Factor out
FactorOut
Factor out
▼
Solve using the Zero Product Property
ZeroProdProp
Use the Zero Product Property
MoveNegDenomToFrac
Put minus sign in front of fraction
CalcQuot
Calculate quotient
Example
The Width of a Path
There is a jogging track around the campground with the same width on every side. It is known that the campground has a rectangular shape with a width of meters and a length of meters.
a Write a trinomial that represents the total area of the campground and the jogging track.
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b The total area for the campground and the jogging track altogether is square meters. What is the width of the jogging track?
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Solution
a Let be the width of the track. The dimensions of the figure can be shown on the diagram.
b To find the width of the jogging track the trinomial found in Part A will be set equal to the total area of the jogging track and campground, square meters.
| Factors of | Sum of Factors |
|---|---|
| and | |
| and | |
| and | |
| and | |
| and | |
| and | |
| and | |
| and | |
| and | |
| and |
▼
Solve using the Zero Product Property
Closure
Finding a Pair of Numbers
The topics learned in this lesson can be applied to the challenge presented at the beginning. Three numbers and are given. The challenge is to find two numbers and that satisfy the following conditions.
If is less than find these two values.
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Hint
Substitute the given numbers and into the equations of the conditions.
Solution
Start by substituting the given numbers and into the equations of the conditions.
In this case, and are factors of and they have a positive sum. To find these numbers, all positive factor pairs of will be listed and their sum will be calculated.
The table shows that and satisfy the conditions. Since is less than and If the given numbers and are the coefficients of a quadratic expression, then this trinomial can be factored by considering the methods and examples already discussed.
First, the Greatest Common Factor of all therms, must be factored out.
The leading coefficient of the trinomial in parentheses is Therefore, it can be factored as where and are factor pairs of whose sum is These values are already known and are and
| Factors of | Sum of Factors |
|---|---|
| and | |
| and |
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