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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to expand (a + b)^2, (a - b)^2, and (a + b)(a - b).

Overview

  • Expanding ( a + b ) 2 and ( a b ) 2
  • Expanding ( a + b ) ( a b )

Three binomial products occur so frequently in algebra that we designate them as special binomial products . We have seen them before (Sections [link] and [link] ), but we will study them again because of their importance as time saving devices and in solving equations (which we will study in a later chapter).

These special products can be shown as the squares of a binomial

( a + b ) 2      and      ( a b ) 2

and as the sum and difference of two terms .

( a + b ) ( a b )

There are two simple rules that allow us to easily expand (multiply out) these binomials. They are well worth memorizing, as they will save a lot of time in the future.

Expanding ( a + b ) 2 And ( a b ) 2

Squaring a binomial

To square a binomial: *

  1. Square the first term.
  2. Take the product of the two terms and double it.
  3. Square the last term.
  4. Add the three results together.

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

Expanding ( a + b ) ( a b )

Sum and difference of two terms

To expand the sum and difference of two terms:

  1. Square the first term and square the second term.
  2. Subtract the square of the second term from the square of the first term.

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


* See problems 56 and 57 at the end of this section.
See problem 58.

Sample set a

( x + 4 ) 2 Square the first term:    x 2 . The product of both terms is 4 x . Double it:    8 x . Square the last term:   16 . Add them together:    x 2 + 8 x + 16. ( x + 4 ) 2 = x 2 + 8 x + 16

Note that ( x + 4 ) 2 x 2 + 4 2 . The 8 x term is missing!

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( a 8 ) 2 Square the first term:    a 2 . The product of both terms is 8 a . Double it:    16 a . Square the last term:    64. Add them together:    a 2 + ( 16 a ) + 64. ( a 8 ) 2 = a 2 16 a + 64

Notice that the sign of the last term in this expression is “ + .” This will always happen since the last term results from a number being squared . Any nonzero number times itself is always positive.

( + ) ( + ) = +    and    ( ) ( ) = +

The sign of the second term in the trinomial will always be the sign that occurs inside the parentheses.

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( y 1 ) 2 Square the first term:    y 2 . The product of both terms is y . Double it:    2 y . Square the last term:    + 1. Add them together:    y 2 + ( 2 y ) + 1.

The square of the binomial 'y minus one' is equal to y squared minus two y plus one. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'minus.' The sign of the last term of the trinomial is labeled as 'plus.'

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( 5 x + 3 ) 2 Square the first term:    25 x 2 . The product of both terms is 15 x . Double it:    30 x . Square the last term:   9 . Add them together:    25 x 2 + 30 x + 9.

The square of the binomial 'five x plus three' is equal to twenty five x squared plus thirty x plus nine. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'plus.' The sign of the last term of the trinomial is also labeled as 'plus.'

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( 7 b 2 ) 2 Square the first term:    49 b 2 . The product of both terms is 14 b . Double it:    28 b . Square the last term:   4 . Add them together:    49 b 2 + ( 28 b ) + 4.

The square of the binomial 'seven b minus two' is equal to forty-nine b squared minus twenty-eight b plus four. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'minus.' The sign of the last term of the trinomial is labeled as 'plus.'

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( x + 6 ) ( x 6 ) Square the first term: x 2 . Subtract the square of the second term ( 36 ) from the square of the first term: x 2 36. ( x + 6 ) ( x 6 ) = x 2 36

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( 4 a 12 ) ( 4 a + 12 ) Square the first term: 16 a 2 . Subtract the square of the second term ( 144 ) from the square of the first term: 16 a 2 144. ( 4 a 12 ) ( 4 a + 12 ) = 16 a 2 144

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( 6 x + 8 y ) ( 6 x 8 y ) Square the first term: 36 x 2 . Subtract the square of the second term ( 64 y 2 ) from the square of the first term: 36 x 2 64 y 2 . ( 6 x + 8 y ) ( 6 x 8 y ) = 36 x 2 64 y 2

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Practice set a

Find the following products.

( x + 5 ) 2

x 2 + 10 x + 25

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( x + 7 ) 2

x 2 + 14 x + 49

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( y 6 ) 2

y 2 12 y + 36

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( 3 a + b ) 2

9 a 2 + 6 a b + b 2

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( 9 m n ) 2

81 m 2 18 m n + n 2

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( 10 x 2 y ) 2

100 x 2 40 x y + 4 y 2

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( 12 a 7 b ) 2

144 a 2 168 a b + 49 b 2

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( 5 h 15 k ) 2

25 h 2 150 h k + 225 k 2

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Exercises

For the following problems, find the products.

( x + 3 ) 2

x 2 + 6 x + 9

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( x + 8 ) 2

x 2 + 16 x + 64

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( y + 9 ) 2

y 2 + 18 y + 81

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( a 4 ) 2

a 2 8 a + 16

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( a 7 ) 2

a 2 14 a + 49

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( b + 15 ) 2

b 2 + 30 b + 225

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( x 12 ) 2

x 2 24 x + 144

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( y 20 ) 2

y 2 40 y + 400

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( 4 x + 2 ) 2

16 x 2 + 16 x + 4

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( 7 x 2 ) 2

49 x 2 28 x + 4

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( 3 a 9 ) 2

9 a 2 54 a + 81

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( 5 a 3 b ) 2

25 a 2 30 a b + 9 b 2

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( 2 h 8 k ) 2

4 h 2 32 h k + 64 k 2

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( a + 1 3 ) 2

a 2 + 2 3 a + 1 9

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( x + 2 5 ) 2

x 2 + 4 5 x + 4 25

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( y 5 6 ) 2

y 2 5 3 y + 25 36

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( x + 1.3 ) 2

x 2 + 2.6 x + 1.69

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( a + 0.5 ) 2

a 2 + a + 0.25

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( x 3.1 ) 2

x 2 6.2 x + 9.61

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( b 0.04 ) 2

b 2 0.08 b + 0.0016

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( x + 5 ) ( x 5 )

x 2 25

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( x + 1 ) ( x 1 )

x 2 1

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( f + 9 ) ( f 9 )

f 2 81

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( 2 y + 3 ) ( 2 y 3 )

4 y 2 9

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( 5 x + 6 ) ( 5 x 6 )

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( 2 a 7 b ) ( 2 a + 7 b )

4 a 2 49 b 2

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( 7 x + 3 t ) ( 7 x 3 t )

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( 5 h 2 k ) ( 5 h + 2 k )

25 h 2 4 k 2

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( x + 1 3 ) ( x 1 3 )

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( a + 2 9 ) ( a 2 9 )

a 2 4 81

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( x + 7 3 ) ( x 7 3 )

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( 2 b + 6 7 ) ( 2 b 6 7 )

4 b 2 36 49

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Expand ( a + b ) 2 to prove it is equal to a 2 + 2 a b + b 2 .

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Expand ( a b ) 2 to prove it is equal to a 2 2 a b + b 2 .

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

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Expand ( a + b ) ( a b ) to prove it is equal to a 2 b 2 .

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Fill in the missing label in the equation below.

The square of the binomial 'a plus b' is equal to a squared plus two ab plus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

first term squared

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Label the parts of the equation below.

The square of the binomial 'a minus b' is equal to a squared minus two ab plus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

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Label the parts of the equation below.

The product of the binomial 'a plus b' and the binomial 'a minus b' is equal to a squared minus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

(a) Square the first term.
(b) Square the second term and subtract it from the first term.

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Exercises for review

( [link] ) Simplify ( x 3 y 0 z 4 ) 5 .

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( [link] ) Find the value of 10 1 2 3 .

1 80

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( [link] ) Find the product. ( x + 6 ) ( x 7 ) .

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( [link] ) Find the product. ( 5 m 3 ) ( 2 m + 3 ) .

10 m 2 + 9 m 9

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( [link] ) Find the product. ( a + 4 ) ( a 2 2 a + 3 ) .

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Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
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Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
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BenJay
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Method
I am eliacin, I need your help in maths
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Amoon
however, may I ask you some questions about Algarba?
Amoon
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Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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