Sat math Essentials

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7 8

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Practice Question
Which of the following are the factors of
z
2
6
z
9?
a.
(
z
3)(
z
3)
b.
(
z
1)(
z
9)
c.
(
z
1)(
z
9)
d.
(
z
3)(
z
3)
e.
(
z
6)(
z
3)
Answer
d.
To find the factors, follow the FOIL method in reverse:
z
2
6
z
9
The product of the 
last
pair of terms equals 
9. There are a few possibilities for these terms: 3 and 3
(because 3 
3 
9),
3 and 
3 (because 
3 
3 
9), 9 and 1 (because 9 
1 
9),
9 and
1 (because 
9 
1 
9).
The sum of the product of the 
outer
pair of terms and the 
inner
pair of terms equals 
6
z
. So we must
choose the two last terms from the list of possibilities that would add up to 
6. The only possibility is
3 and 
3. Therefore, we know the last terms are 
3 and 
3.
The product of the 
first
pair of terms equals 
z
2
. The most likely two terms for the first pair is 
z
and 
z
because 
z
z
z
2
.
Therefore, the factors are (
z
3)(
z
3).
Fractions with Variables
You can work with fractions with variables the same as you would work with fractions without variables.
Example
Write 
6
x
1
x
2
as a single fraction.
First determine the LCD of 6 and 12: The LCD is 12. Then convert each fraction into an equivalent fraction
with 12 as the denominator:
6
x
1
x
2
6
x
2
2
1
x
2
1
2
2
x
1
x
2
Then simplify:
1
2
2
x
1
x
2
1
x
2
Practice Question
Which of the following best simplifies 
5
8
x
2
5
x
?
a.
4
9
0
b.
4
9
0
x
c.
5
x
d.
4
3
0
x
e.
x
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7 9

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Answer
b.
To simplify the expression, first determine the LCD of 8 and 5: The LCD is 40. Then convert each frac-
tion into an equivalent fraction with 40 as the denominator:
5
8
x
2
5
x
(5
x
5
8
5) 
(
(
2
5
x
8
8
)
)
2
4
5
0
x
1
4
6
0
x
Then simplify:
2
4
5
0
x
1
4
6
0
x
4
9
0
x
Reciprocal Rules
There are special rules for the sum and difference of reciprocals. The following formulas can be memorized for
the SAT to save time when answering questions about reciprocals:
■
If
x
and 
y
are not 0, then 
1
x
y
x
x
y
y
■
If
x
and 
y
are not 0, then 
1
x
1
y
y
x
y
x
Note:
These rules are easy to figure out using the techniques of the last section, if you are comfortable with
them and don’t like having too many formulas to memorize.
Quadratic Equations
A 
quadratic equation
is an equation in the form 
ax
2
bx
c
0, where 
a
,
b
, and 
c
are numbers and 
a
≠
0. For
example,
x
2
6
x
10 
0 and 6
x
2
8
x
22 
0 are quadratic equations.
Zero-Product Rule 
Because quadratic equations can be written as an expression equal to zero, the zero-product rule is useful when
solving these equations.
The 
zero-product rule
states that if the product of two or more numbers is 0, then at least one of the num-
bers is 0. In other words, if
ab
0, then you know that either 
a
or 
b
equals zero (or they both might be zero). This
idea also applies when 
a
and 
b
are factors of an equation. When an equation equals 0, you know that one of the
factors of the equation must equal zero, so you can determine the two possible values of
x
that make the factors
equal to zero.
Example
Find the two possible values of
x
that make this equation true
:
(
x
4)(
x
2) 
0
Using the zero-product rule, you know that either 
x
4 
0 or that 
x
2 
0.
So solve both of these possible equations:
x
4 
0
x
2 
0
x
4 
4 
0 
4
x
2 
2 
0 
2
x
4
x
2
Thus, you know that both 
x
4 and 
x
2 will make (
x
4)(
x
2) 
0.
The zero product rule is useful when solving quadratic equations because you can rewrite a quadratic equa-
tion as equal to zero and take advantage of the fact that one of the factors of the quadratic equation is thus equal
to 0.
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8 0

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Practice Question
If (
x
8)(
x
5) 
0, what are the two possible values of
x
?
a.
x
8 and 
x
5
b.
x
8 and 
x
5
c.
x
8 and 
x
0
d.
x
0 and 
x
5
e.
x
13 and 
x
13
Answer
a.
If (
x
8)(
x
5) 
0, then one (or both) of the factors must equal 0.
x
8 
0 if
x
8 because 8 
8 
0.
x
5 
0 if
x
5 because 
5 
5 
0.
Therefore, the two values of
x
that make (
x
8)(
x
5) 
0 are
x
8 and
x
5.
Solving Quadratic Equations by Factoring
If a quadratic equation is not equal to zero, rewrite it so that you can solve it using the zero-product rule.
Example
If you need to solve 
x
2
11
x
12, subtract 12 from both sides:
x
2
11
x
12 
12 
12
x
2
11
x
12 
0
Now this quadratic equation can be solved using the zero-product rule:
x
2
11
x
12 
0
(
x
12)(
x
1) 
0
Therefore:
x
12 
0
or
x
1 
0
x
12 
12 
0 
12
x
1 
1 
0 
1
x
12
x
1
Thus, you know that both 
x
12 and 
x
1 will make 
x
2
11
x
12 
0.
A quadratic equation must be factored before using the zero-product rule to solve it.
Example
To solve 
x
2
9
x
0, first factor it:
x
(
x
9) 
0.
Now you can solve it.
Either 
x
0 or 
x
9 
0.
Therefore, possible solutions are 
x
0 and 
x
9.
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8 1

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Practice Question
If
x
2
8
x
20, which of the following could be a value of
x
2
8
x
?
a.
20
b.
20
c.
28
d.
108
e.
180
Answer
e.
This question requires several steps to answer. First, you must determine the possible values of
x
con-
sidering that 
x
2
8
x
20. To find the possible 
x
values, rewrite 
x
2
8
x
20 as 
x
2
8
x
20 
0, fac-
tor, and then use the zero-product rule.
x
2
8
x
20 
0 is factored as (
x
10)(
x
2).
Thus, possible values of
x
are 
x
10 and 
x
2 because 10 
10 
0 and 
2 
2 
0.
Now, to find possible values of
x
2
8
x
, plug in the
x
values:
If
x
2, then 
x
2
8
x
(
2)
2
(8)(
2) 
4 
(
16) 
12. None of the answer choices is
12, so try 
x
10.
If
x
10, then 
x
2
8
x
10
2
(8)(10) 
100 
80 
180.
Therefore, answer choice 
e
is correct.
G r a p h s o f Q u a d r a t i c E q u a t i o n s
The (
x
,
y
) solutions to quadratic equations can be plotted on a graph. It is important to be able to look at an equa-
tion and understand what its graph will look like. You must be able to determine what calculation to perform on
each 
x
value to produce its corresponding 
y
value.
For example, below is the graph of
y
x
2
.
The equation 
y
x
2
tells you that for every 
x
value, you must square the 
x
value to find its corresponding 
y
value. Let’s explore the graph with a few 
x
-coordinates:
An 
x
value of 1 produces what 
y
value? Plug 
x
1 into 
y
x
2
.
x
1
2
3
4
5
6
7
1
2
3
4
5
–1
–2
–3
–1
–2
–3
–4
–5
–6
–7
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8 2

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When 
x
1,
y
1
2
, so 
y
1.
Therefore, you know a coordinate in the graph of
y
x
2
is (1,1).
An 
x
value of 2 produces what 
y
value? Plug 
x
2 into 
y
x
2
.
When 
x
2,
y
2
2
, so 
y
4.
Therefore, you know a coordinate in the graph of
y
x
2
is (2,4).
An 
x
value of 3 produces what 
y
value? Plug 
x
3 into 
y
x
2
.
When 
x
3,
y
3
2
, so 
y
9.
Therefore, you know a coordinate in the graph of
y
x
2
is (3,9).
The SAT may ask you, for example, to compare the graph of
y
x
2
with the graph of
y
(
x
1)
2
. Let’s com-
pare what happens when you plug numbers (
x
values) into 
y
(
x
1)
2
with what happens when you plug num-
bers (
x
values) into 
y
x
2
:
y
= 
x
2
y
= (
x
1)
2
If
x
= 1,
y
= 1.
If
x
= 1,
y
= 0.
If
x
= 2,
y
= 4.
If
x
= 2,
y
= 1.
If
x
= 3,
y
= 9.
If
x
= 3,
y
= 4.
If
x
= 4,
y
= 16.
If
x
= 4,
y
= 9.
The two equations have the same 
y
values, but they match up with different 
x
values because 
y
(
x
1)
2
subtracts 1 before squaring the 
x
value. As a result, the graph of
y
(
x
1)
2
looks identical to the graph of
y 
x
2
except that the base is shifted to the right (on the 
x
-axis) by 1:
How would the graph of
y
x
2
compare with the graph of
y
x
2
1?
In order to find a 
y
value with 
y
x
2
, you square the 
x
value. In order to find a 
y
value with 
y
x
2
1, you
square the 
x
value and then subtract 1. This means the graph of
y
x
2
1 looks identical to the graph of
y 
x
2
except that the base is shifted down (on the 
y
-axis) by 1:
x
y
1
2
3
4
5
6
7
1
2
3
4
5
–1
–2
–3
–1
–2
–3
–4
–5
–6
–7