The following three formulae should be memorized.
x
+
a
2
=
x
2
+
2
ax
+
a
2
x
+
a
2
=
x
2
+
2
ax
+
a
2
size 12{ left (x+a right ) rSup { size 8{2} } =x rSup { size 8{2} } +2 ital "ax"+a rSup { size 8{2} } } {}
(1)
x
−
a
2
=
x
2
−
2
ax
+
a
2
x
−
a
2
=
x
2
−
2
ax
+
a
2
size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {}
(2)
x
+
a
x
−
a
=
x
2
−
a
2
x
+
a
x
−
a
=
x
2
−
a
2
size 12{ left (x+a right ) left (x - a right )=x rSup { size 8{2} } - a rSup { size 8{2} } } {}
(3)It is important to have these three formulae on the top of your head. It is also nice to be able to show why these formulae work, for instance by using FOIL. But the most important thing of all is knowing what these three formulae mean, and how to use them.
These three are all “algebraic generalizations,” as discussed in the first unit on functions. That is, they are equations that hold true for any values of
xx size 12{x} {} and
aa size 12{a} {}. It may help if you think of the second equation above as standing for:
Anthing−Anything Else2=Anything2−2Anything Else2Anthing−Anything Else2=Anything2−2Anything Else2 size 12{ left ("Anthing" - "Anything Else" right ) rSup { size 8{2} } ="Anything" rSup { size 8{2} } - 2 left ("Anything Else" right ) rSup { size 8{2} } } {}For instance, suppose the Anything (or
xx size 12{x} {}) is 5, and the Anything Else (or
aa size 12{a} {}) is 3.
x−a2=x2−2ax+a2x−a2=x2−2ax+a2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {},
when
x=5x=5 size 12{x=5} {},
a=3a=3 size 12{a=3} {}.
-
5
−
3
2
=
?
5
2
−
2
3
5
+
3
2
5
−
3
2
=
?
5
2
−
2
3
5
+
3
2
size 12{5 - 3 rSup { size 8{2} } { {}={}} cSup { size 8{?} } 5 rSup { size 8{2} } - 2 left (3 right ) left (5 right )+3 rSup { size 8{2} } } {}
-
2
2
=
?
25
−
30
+
9
2
2
=
?
25
−
30
+
9
size 12{2 rSup { size 8{2} } { {}={}} cSup { size 8{?} } "25" - "30"+9} {}
-
4
=
4
4
=
4
size 12{4=4} {}
It worked! Now, let’s leave the Anything as
xx size 12{x} {}, but play with different values of
aa size 12{a} {}.
More examples of
x−a2=x2−2ax+a2x−a2=x2−2ax+a2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {}
a
=
1
:
(
x
-
1
)
2
=
x
2
-
2
x
+
1
a
=
2
:
(
x
-
2
)
2
=
x
2
-
4
x
+
4
a
=
3
:
(
x
-
3
)
2
=
x
2
-
6
x
+
9
a
=
5
:
(
x
-
5
)
2
=
x
2
-
10
x
+
25
a
=
10
:
(
x
-
10
)
2
=
x
2
-
20
x
+
100
a
=
1
:
(
x
-
1
)
2
=
x
2
-
2
x
+
1
a
=
2
:
(
x
-
2
)
2
=
x
2
-
4
x
+
4
a
=
3
:
(
x
-
3
)
2
=
x
2
-
6
x
+
9
a
=
5
:
(
x
-
5
)
2
=
x
2
-
10
x
+
25
a
=
10
:
(
x
-
10
)
2
=
x
2
-
20
x
+
100
(4)Once you’ve seen a few of these, the pattern becomes evident: the number doubles to create the middle term (the coefficient of
xx size 12{x} {}), and squares to create the final term (the number).
The hardest thing about this formula is remembering to use it. For instance, suppose you are asked to expand:
2y
−
6
2
2y
−
6
2
size 12{ left (2y - 6 right ) rSup { size 8{2} } } {}
(5)There are three ways you can approach this.
Table 1
|
2y
−
6
2
2y
−
6
2
, computed three different ways
|
| Square each term |
FOIL |
Using the formula above |
|
2y
−
6
2
2y
2
−
2
6
2y
+
6
2
4y
2
−
24
y
+
36
2y
−
6
2
2y
2
−
2
6
2y
+
6
2
4y
2
−
24
y
+
36
size 12{ matrix {
{} # left (2y - 6 right ) rSup { size 8{2} } {} ##
={} {} # left (2y right ) rSup { size 8{2} } - 2 left (6 right ) left (2y right )+6 rSup { size 8{2} } {} ##
={} {} # 4y rSup { size 8{2} } - "24"y+"36"{}
} } {}
|
2y
−
6
2y
−
6
2y
2y
−
2y
6
−
2y
6
+
36
4y
2
−
12
y
−
12
y
+
36
4y
2
−
24
y
+
36
2y
−
6
2y
−
6
2y
2y
−
2y
6
−
2y
6
+
36
4y
2
−
12
y
−
12
y
+
36
4y
2
−
24
y
+
36
size 12{ matrix {
{} # left (2y - 6 right ) left (2y - 6 right ) {} ##
={} {} # left (2y right ) left (2y right ) - left (2y right )6 - left (2y right )6+"36" {} ##
={} {} # 4y rSup { size 8{2} } - "12"y - "12"y+"36" {} ##
={} {} # 4y rSup { size 8{2} } - "24"y+"36"{}
} } {}
|
2y
−
6
2
2y
2
−
2
6
2y
+
6
2
4y
2
−
24
y
+
36
2y
−
6
2
2y
2
−
2
6
2y
+
6
2
4y
2
−
24
y
+
36
size 12{ matrix {
{} # left (2y - 6 right ) rSup { size 8{2} } {} ##
={} {} # left (2y right ) rSup { size 8{2} } - 2 left (6 right ) left (2y right )+6 rSup { size 8{2} } {} ##
={} {} # 4y rSup { size 8{2} } - "24"y+"36"{}
} } {}
|
Did it work? If a formula is true, it should work for any
y
y
size 12{y} {}
-value; let’s test each one with
y
=
5
y
=
5
size 12{y=5} {}
. (Note that the second two methods got the same answer, so we only need to test that once.)
Table 2
|
2y
−
6
2
=
?
4y
2
−
36
2
⋅
5
−
6
2
=
?
4y
2
−
36
10
−
6
2
=
?
100
−
36
4
2
=
?
64
✗
2y
−
6
2
=
?
4y
2
−
36
2
⋅
5
−
6
2
=
?
4y
2
−
36
10
−
6
2
=
?
100
−
36
4
2
=
?
64
✗
size 12{ matrix {
left (2y - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "36" {} ##
left (2 cdot 5 - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "36" {} ##
left ("10" - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "100" - "36" {} ##
4 rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "64"{}
} } {}
|
2y
−
6
2
=
?
4y
2
−
24
y
+
36
2
⋅
5
−
6
2
=
?
4
5
2
−
24
⋅
5
+
36
10
−
6
2
=
?
100
−
120
+
36
4
2
=
?
16
✓
2y
−
6
2
=
?
4y
2
−
24
y
+
36
2
⋅
5
−
6
2
=
?
4
5
2
−
24
⋅
5
+
36
10
−
6
2
=
?
100
−
120
+
36
4
2
=
?
16
✓
size 12{ matrix {
left (2y - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "24"y+"36" {} ##
left (2 cdot 5 - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4 left (5 right ) rSup { size 8{2} } - "24" cdot 5+"36" {} ##
left ("10" - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "100" - "120"+"36" {} ##
4 rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "16"{}
} } {}
|
We conclude that squaring each term individually does not work. The other two methods both give the same answer, which works.
The first method is the easiest, of course. And it looks good.
2y22y2 size 12{ left (2y right ) rSup { size 8{2} } } {} is indeed
4y24y2 size 12{4y rSup { size 8{2} } } {}. And
6262 size 12{6 rSup { size 8{2} } } {} is indeed 36. But as you can see, it led us to a false answer—an algebraic generalization that did not hold up.
I just can’t stress this point enough. It sounds like a detail, but it causes errors all through Algebra II and beyond. When you’re adding or subtracting things, and then squaring them, you can’t just square them one at a time. Mathematically,
x+a2≠x2+a2x+a2≠x2+a2 size 12{ left (x+a right ) rSup { size 8{2} } <> x rSup { size 8{2} } +a rSup { size 8{2} } } {}. You can confirm this with numbers all day.
7+32=1007+32=100 size 12{ left (7+3 right ) rSup { size 8{2} } ="100"} {}, but
72+32=5872+32=58 size 12{7 rSup { size 8{2} } +3 rSup { size 8{2} } ="58"} {}. They’re not the same.
So that leaves the other two methods. FOIL will never lead you astray. But the third approach, the formula, has three distinct advantages.
- The formula is faster than FOIL.
- Using these formulae is a specific case of the vital mathematical skill of using any formula—learning how to plug numbers and variables into some equation that you’ve been given, and therefore understanding the abstraction that formulae represent.
- Before this unit is done, we will be completing the square, which requires running that particular formula backward—which you cannot do with FOIL.
Multiplying Binomials
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