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Algebraic Formula Sheet
Arithmetic Operations ac + bc = c(a + b)
( a b
c
a bc
a b
c d
ad + bc bd
a − b c − d
b − a d − c
ab + ac a = b + c, a 6 = 0
a
b c
ab c
(a b c
) (^) = ac b
a c
c d
ad − bc bd
a + b c
a c
b c ( a b
c d
) (^) = ad bc
Properties of Exponents xnxm^ = xn+m
(xn)m^ = xnm
(xy)n^ = xnyn
x mn =
x m^1 )n =
xn
) (^) m 1
x y
)−n
y x
)n
yn xn
x^0 = 1, x 6 = 0
( x y
)n
xn yn
1 x−n^ = xn
xn xm^
= xn−m
x−n^ =
xn
Properties of Radicals √ nx = x n 1
√nxy = √nx √ny
m
√nx = mn √x
n
x y
√ nx √ ny
√ nxn (^) = x, if n is odd
√ nxn (^) = |x|, if n is even
Properties of Inequalities
If a < b then a + c < b + c and a − c < b − c
If a < b and c > 0 then ac < bc and a c
b c If a < b and c < 0 then ac > bc and a c
b c
Properties of Absolute Value
|x| =
x if x ≥ 0 −x if x < 0
|x| ≥ 0
|xy| = |x||y|
| − x| = |x|
x y
∣ =^
|x| |y|
|x + y| ≤ |x| + |y| Triangle Inequality
|x − y| ≥
∣|x| − |y|
∣ Reverse Triangle Inequality
Distance Formula Given two points, PA = (x 1 , y 1 ) and PB = (x 2 , y 2 ), the distance between the two can be found by:
d(PA, PB ) =
(x 2 − x 1 )^2 + (y 2 − y 1 )^2
Number Classifications Natural Numbers : N={1, 2, 3, 4, 5,.. .}
Whole Numbers : {0, 1, 2, 3, 4, 5,.. .}
Integers : Z={... ,-3, -2, -1, 0, 1, 2, 3, .. .}
Rationals : Q=
All numbers that can be writ- ten as a fraction with an integer numerator and a nonzero integer denominator, a b
Irrationals : {All numbers that cannot be ex- pressed as the ratio of two integers, for example√ 5,
27, and π}
Real Numbers : R={All numbers that are either a rational or an irrational number}
Logarithms and Log Properties
Definition
y = logb x is equivalent to x = by
Example
log 2 16 = 4 because 24 = 16
Special Logarithms
ln x = loge x natural log where e=2.718281828...
log x = log 10 x common log
Logarithm Properties
logb b = 1
logb bx^ = x
ln ex^ = x
logb 1 = 0
blogb^ x^ = x
eln^ x^ = x
logb (xk) = k logb x
logb (xy) = logb x + logb y
logb
x y
= logb x − logb y
Factoring
xa + xb = x(a + b)
x^2 − y^2 = (x + y)(x − y)
x^2 + 2xy + y^2 = (x + y)^2
x^2 − 2 xy + y^2 = (x − y)^2
x^3 + 3x^2 y + 3xy^2 + y^3 = (x + y)^3
x^3 − 3 x^2 y + 3xy^2 − y^3 = (x − y)^3
x^3 + y^3 = (x + y)
x^2 − xy + y^2
x^3 − y^3 = (x − y)
x^2 + xy + y^2
x^2 n^ − y^2 n^ = (xn^ − yn) (xn^ + yn)
If n is odd then,
xn^ − yn^ = (x − y)
xn−^1 + xn−^2 y + ... + yn−^1
xn^ + yn^ = (x + y)
xn−^1 − xn−^2 y + xn−^3 y^2 ... − yn−^1
Linear Functions and Formulas
Examples of Linear Functions
x
y
y = x
linear f unction
x
y
y = 1
constant f unction
Quadratics and Solving for x
Quadratic Formula To solve ax^2 + bx + c = 0, a 6 = 0, use :
x = −b ±
b^2 − 4 ac 2 a
The Discriminant The discriminant is the part of the quadratic equation under the radical, b^2 − 4 ac. We use the discriminant to determine the number of real solutions of ax^2 + bx + c = 0 as such :
- If b^2 − 4 ac > 0, there are two real solutions.
- If b^2 − 4 ac = 0, there is one real solution.
- If b^2 − 4 ac < 0, there are no real solutions.
Square Root Property
Let k be a nonnegative number. Then the solutions to the equation
x^2 = k
are given by x = ±
k.
Other Useful Formulas
Compound Interest
A = P
r n
)nt
where: P= principal of P dollars r= Interest rate (expressed in decimal form) n= number of times compounded per year t= time
Continuously Compounded Interest
A = P ert
where: P= principal of P dollars r= Interest rate (expressed in decimal form) t= time
Circle
(x − h)^2 + (y − k)^2 = r^2
This graph is a circle with radius r and center (h, k).
Ellipse
(x − h)^2 a^2
(y − k)^2 b^2
This graph is an ellipse with center (h, k) with vertices a units right/left from the center and vertices b units up/down from the center.
Hyperbola
(x − h)^2 a^2
(y − k)^2 b^2
This graph is a hyperbola that opens left and right, has center (h, k), vertices (h ± a, k); foci (h ± c, k), where c comes from c^2 = a^2 + b^2 and asymptotes that pass through the center y = ± b a
(x − h) + k.
(y − k)^2 a^2
(x − h)^2 b^2
This graph is a hyperbola that opens up and down, has center (h, k), vertices (h, k ± a); foci (h, k ± c), where c comes from c^2 = a^2 + b^2 and asymptotes that pass through the center y = ± a b (x − h) + k.
Pythagorean Theorem
A triangle with legs a and b and hypotenuse c is a right triangle if and only if
a^2 + b^2 = c^2
