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We’ll work one now where the middle term is negative.

Factor: 81 y 2 72 y + 16 .

Solution

The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be ( a b ) 2 .

.
Are the first and last terms perfect squares? .
Check the middle term. .
Does is match ( a b ) 2 ? Yes. .
Write the square of a binomial. .
Check by mulitplying.
( 9 y 4 ) 2
( 9 y ) 2 2 9 y 4 + 4 2
81 y 2 72 y + 16

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Factor: 64 y 2 80 y + 25 .

( 8 y 5 ) 2

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Factor: 16 z 2 72 z + 81 .

( 4 z 9 ) 2

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The next example will be a perfect square trinomial with two variables.

Factor: 36 x 2 + 84 x y + 49 y 2 .

Solution

.
Test each term to verify the pattern. .
Factor. .
Check by mulitplying.
( 6 x + 7 y ) 2
( 6 x ) 2 + 2 6 x 7 y + ( 7 y ) 2
36 x 2 + 84 x y + 49 y 2

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Factor: 49 x 2 + 84 x y + 36 y 2 .

( 7 x + 6 y ) 2

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Factor: 64 m 2 + 112 m n + 49 n 2 .

( 8 m + 7 n ) 2

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Factor: 9 x 2 + 50 x + 25 .

Solution

9 x 2 + 50 x + 25 Are the first and last terms perfect squares? ( 3 x ) 2 ( 5 ) 2 Check the middle term—is it 2 a b ? ( 3 x ) 2 2 ( 3 x ) ( 5 ) 30 x ( 5 ) 2 No! 30 x 50 x This does not fit the pattern! Factor using the “ac” method. 9 x 2 + 50 x + 25 Notice: a c 9 · 25 225 and 5 · 45 = 225 5 + 45 = 50 Split the middle term. 9 x 2 + 5 x + 45 x + 25 Factor by grouping. x ( 9 x + 5 ) + 5 ( 9 x + 5 ) ( 9 x + 5 ) ( x + 5 ) Check. ( 9 x + 5 ) ( x + 5 ) 9 x 2 + 45 x + 5 x + 25 9 x 2 + 50 x + 25

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Factor: 16 r 2 + 30 r s + 9 s 2 .

( 8 r + 3 s ) ( 2 r + 3 s )

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Factor: 9 u 2 + 87 u + 100 .

( 3 u + 4 ) ( 3 u + 25 )

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Remember the very first step in our Strategy for Factoring Polynomials? It was to ask “is there a greatest common factor?” and, if there was, you factor the GCF before going any further. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.

Factor: 36 x 2 y 48 x y + 16 y .

Solution

36 x 2 y 48 x y + 16 y
Is there a GCF? Yes, 4 y , so factor it out. 4 y ( 9 x 2 12 x + 4 )
Is this a perfect square trinomial?
Verify the pattern. .
Factor. 4 y ( 3 x 2 ) 2
Remember: Keep the factor 4 y in the final product.
Check.
4 y ( 3 x 2 ) 2
4 y [ ( 3 x ) 2 2 · 3 x · 2 + 2 2 ]
4 y ( 9 x ) 2 12 x + 4
36 x 2 y 48 x y + 16 y

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Factor: 8 x 2 y 24 x y + 18 y .

2 y ( 2 x 3 ) 2

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Factor: 27 p 2 q + 90 p q + 75 q .

3 q ( 3 p + 5 ) 2

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Factor differences of squares

The other special product you saw in the previous was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

( 3 x 4 ) ( 3 x + 4 ) 9 x 2 16

Remember, when you multiply conjugate binomials, the middle terms of the product add to 0. All you have left is a binomial, the difference of squares.

Multiplying conjugates is the only way to get a binomial from the product of two binomials.

Product of conjugates pattern

If a and b are real numbers

( a b ) ( a + b ) = a 2 b 2

The product is called a difference of squares.

To factor, we will use the product pattern “in reverse” to factor the difference of squares. A difference of squares factors to a product of conjugates.

Difference of squares pattern

If a and b are real numbers,

This image shows the difference of two squares formula, a squared – b squared = (a – b)(a + b). Also, the squares are labeled, a squared and b squared. The difference is shown between the two terms. Finally, the factoring (a – b)(a + b) are labeled as conjugates.

Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.

How to factor differences of squares

Factor: x 2 4 .

Solution

This table gives the steps for factoring x squared minus 4. The first step is identifying the pattern in the binomial including it is a difference. Also, the first and last terms are verified as squares. The second step is writing the two terms as squares, x squared and 2 squared. The second step is writing the two terms as squares, x squared and 2 squared. The third step is to write the factoring as a product of the conjugates (x – 2)(x + 2). The last step is to check with multiplication.
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Factor: h 2 81 .

( h 9 ) ( h + 9 )

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Factor: k 2 121 .

( k 11 ) ( k + 11 )

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Factor differences of squares.

Step 1. Does the binomial fit the pattern? a 2 b 2 Is this a difference? ____ ____ Are the first and last terms perfect squares? Step 2. Write them as squares. ( a ) 2 ( b ) 2 Step 3. Write the product of conjugates. ( a b ) ( a + b ) Step 4. Check by multiplying.

Questions & Answers

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
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The Reply
Felecia answer 1.5 hours before he reaches her
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I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
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Christian
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gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
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Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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