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0.2 Equations
Consistency of Variables: In math, the same variables (both our alphabet and the Greek alphabet) are
generally used to show certain specific things.
Ex. Variables in Equations to be Graphed ________________________
Ex. Constant Values _____________________
An equation might be true for some values of the variables involved and false for others.
1) Contains a finite number of points
x2 – 1 = 0
2) Is true for all values
x2 + 2x + 1 = (x + 1)2
3) Contains an infinite number of points
y = x2 – 1
4) Has no real solution
x2 = -1
Factoring might be necessary to find solutions:
*___________________________ and factor!
Why do we do this? AB = if and only if A = 0 or B = 0.
Ex) x 2  3 x  4
Ex) x 2  2 x  1
Quadratic Formula can find x-intercepts:
Ex) 3 x 2  7 x  1
Other factoring patterns:
a2 – b2 = (a – b)(a + b)
a3 – b3 = (a – b)(a2 + ab + b2)
an – bn = (a – b)(an-1 + an-2b + an-3b2 + an-4b3 + …. + a2bn-3 + abn-2 + bn-1)
Ex) Factor a5 – b5
Ex) 27-64x3
6 Fraction Rules:
a c ad  bc
 
when b  0, d  0
b d
bd
a c ac
 
when b  0, d  0
b d bd
a
b  a  d  ad when b  0, c  0, d  0
c b c bc
d
 a  ca
c  
when b  0
b b
a
a
ac

when b  0, c  0
b
b
c
b
c

a
when b  0, c  0
bc
x2 1
Ex) Simplify
x 1
A quotient equals 0 when:
its _______________________ equals 0 and the ______________________ does not equal zero.
x2 1
0
Ex) Solve
x 1
Ex) Solve
x 1
4
x 1
x3  x 2  6 x
0
Ex) Solve 2
x  3x  2
3
1

Ex) Simplify x  2 x
x 1
Systems of Equations: Solve to find the solution(s) that would be true for BOTH/ALL equations.
 x 2  1  0
Ex)  2
 x  x  2  0
 x2  y  3
Ex) 
2 x  y  0
x  2 y  z  0

Ex.  x  y  z  1
x  2 y  z  2

*We will skip this as it is not used often in calculus. If you would like to follow a similar problem for your
own interest, look on page 21 ex. 4.
Homework: 7, 11-37(odd) we did 27 in class, 43, 53-63(odd), 69, 79