Algebra Basics Flashcards
topic
manipulating equations
- solving for one variable
- solving equations with fractions
- solving for two or more variables
topic
Simulatneous equations
given two equations, solve for the value of an expression as opposed to the value of the variables
topic
inequalities
statements much like those with = signs. can be manipulated by the same rules, with the notable expection of flipping the direction of the < or > when you multiple or divide by a negative number
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quadratic equations
- expanded quadratic equations
- fully factored quadratic equations
- FOIL
- common quadratic equations
topic
exponents
- multiplying exponents
- dividing exponents
- exponents raised to a power
- negative exponents
- zero exponents
- factoring exponents
topic
square roots
- adding and subtracting square roots
- multiplying and dividing square roots
- simplifying square roots
golden rule of manipulating equations
whatever you do to one side of the equation, you must do to the other side of the equation
operation pairs
addition– subtraction
division– multiplication
power– square root
solving equations with fractions
- clear the fraction by multiplying the fraction by the value in the denominator
- perform the same operation (multiplication by whatever the denominator was) to each other term in the expression
equations with more than one variable
- given two equations, find a way to match at least one of the coefficients by multiplying or dividing
- subtract the newly formed equation from the first original equation to find one of the variable values
- plug in the variable value into one of the original equations to solve for the other variable
simultaneous equations
apply same method as two variable equations, end up with an expression instead of a value for the variables
combining ranges
consider all possible combinations of x and y (and then all possible combinations of x and y in the defined expression)
Calculate: (example x-y)
1. the greatest x minus the greatest y
2. the greatest x minus the least y
3. the least x minus the greatest y
4. the least x minus the least y
the greatest value from the above calculation and the least consitutes the range of x-y
fully expanded quadratic equations
example: x^2 + 10x + 24
contains a squared variable, a variable multiplied by some number, and an additional number
factoring an expanded quadratic
goal: break the equation apart to find the values for the variables that make the equation true. these are referred to as the roots of the equation, and the roots of a quadratic equation are the variables that make the expression equal to 0
steps to factoring
- seperate the x^2 into two cvalues of x and place them inside their own parantheses
- consider the second and third terms in the equation and find the factor pair of the third term that, when added or subtracted, yields the second term
- determine the operations that are inside the parentheses
- once the parentheses have been filled in, solve for the roots of the equation
fully factored quadratic equations
in order to expand a fully factored quadratic equation into the fully expanded standard form, employ the FOIL tactic
difference of squares
x^2-y^2=(x+y)(x-y)
common quadratics
- difference of squares
- (x+y)2=x^2+2xy+y^2
- (x-y)2=x^2-2xy+y^2
exponents
shorthand for repeated multiplation, composed of a base and a “power”. The power tells you how many times to multiply the base by itself
MADSPM
First three rules of exponents.
1. Multiply– Add
2. Divide– Subtract
3. Power– Multiply
negative exponent
any term raised to a negative power is equal to the reciprocal of that term raised to the positive power. 8^-2 = 1/8^2 = 1/64
zero exponent
any nonzero number raised to apower of 0 is equal to 1.
exponent tip 1:
if none of the bases in an exponent equation match up, search for a way to rewrite the bases so they do match
exponent tip 2:
if you get stuck on an exponent question, try to factor the expression to produce values that can be manipulated by one of the five rules
base greater than 1 to a power greater than one
always results in a greater number
base between 0 and 1 to a power greater than 1
results in a smaller number
negative number raised to an even power
results in a positive number
negative number raised to an odd power
results ina negative number
a number raised to the first power
always results in the number itself
square root
a square root of a number, x, is the number, y, for which y^2 = x. taking the square root is the opposite of finding the square of a number. the square root of a number is always non-negative
adding and subtracting square roots
add or subtract only if the values under the radical sign are equal
multiplying and dividing square roots
any square root can be multiplied or divided– no restrictions. multiplying square roots is as simple as multiplying the terms outside square roots together and the terms underneath the radical signs togather
simplifying square roots
simplify a square root by looking for ways to factor the number under the root that results in at least one perfect square
