Algebra A Cheat Sheet

Algebra Cheat Sheet, Roth 5/2007 This set of notes is organized accordingly: 1. Number Systems 5.5 ||, _|_ , H & V Lines 2.5 Order of Operations 5.6 Point Slope Form 3. Number Operations 5.7 Standard Form 4.1 Variables 5.8 Linear Inequalities 4.2 Expressions 5.9 Absolute Value 4.3 Equations 7.1 Systems by Graph 4.4 MultiStep Equations 7.2 Syst by Substitution 4.5 Ratio, Proportion, % 7.3 Syst by Elimination 4.6 Inequalities 7.4 Syst of Inequalities 5.1 Two Variable Eqns 11 Basic Geometry 5.2 Tables & Graphs 14 Probability 5.3 Function & Relation 3.5 Exponents 5.4 Slope Intercept 6.1 Poly. & Quadratics 6.3 Rational Exp & Eqn 6.6 Radicals See the accompanying Math Fundamentals sheet for prerequisite concepts of negative & positive number operations, fractions, etc,. The Real Numbers – All numbers (the set of both Rationals and Irrationals) Properties of Numbers The following formal names for the properties of numbers for + & x may appear on tests. They look strange in their abstract definitions, but we ‘know’ what they mean by common sense.         Closure: a + b & a*b are real Commutative: a + b = b + a, ab = ba Associative: a + (b + c) = (a + b) + c & a(bc) = (ab)c Identity: Inverse: a+0=a & a*1= a a + (-a) = 0 & a * 1/a = 1 Distributive: a(b + c) = ab + ac (ref. area model) Multiplication of 0: a * 0 = 0 Multiplicative of (-1): w * (-1) = -w 2.4 Decimals Place value is a fundamental concept for our Base 10 numbering system & decimals extend the place value concept to numbers less than one. Consider a number like 1234.56. It is: 1 x 1000 = 1000 2 x 100 = 200 3 x 10 = 30 4x1 = 4 5 x 1/10 = .5 6 x 1/100 = .06 added up 1234.56 1. Number Systems Types of Numbers The types of numbers build up from the most basic (counting numbers) to the most complete types of numbers like Real’s. As we move down the list, the simpler types of numbers are also included (A Natural Number like 2 is also a Whole Number, Integer, Rational, and a Real's) Natural Numbers - Counting numbers: {1, 2, 3, 4, } Whole Numbers - Natural plus 0: { 0, 1, 2, 3, 4, .. .} Integers - Whole plus negatives:{.., –2, –1, 0, 1, 2, ..} Rational Numbers - All numbers that can be expressed as a "ratio" 2.5 Order of Operations Remember the acrostic PEMDAS – Please Excuse My Dear Aunt Sally. P Do any operations within Parentheses, like ( ), [ ], or { }, first. E Exponents like 32 and x3 M Multiplication and Division, left to right. Left-toright order does not matter if only multiplication is involved, but it matters for division D As above A Addition and Subtraction, left to right. S As above a where a & b are integers b (but b cannot be zero). We usually call them fractions. The bottom of the fraction is called the denominator and the top the numerator. Terminating decimals (they end - don't go on forever) are also rational numbers. – i.e 4.124589 Repeating decimals (they have a block of the same numbers that repeat forever). 2.565656…. Irrational Numbers – A number that cannot be expressed as a ratio of integers. As decimals they never repeat or terminate. Radicals √2, √3, √5, & special numbers like π & e are common examples. 3. Number Operations Absolute Value | -2 | Absolute value gives the positive magnitude of a number – the distance from zero whether + or -. In practice it makes everything within it positive. | 34 | = 34 | -25 | = 25 Algebra A Cheat Sheet | 2 – 34 | = | -32 | = 32 Exponents base (exponent) Coefficient: The number multiplying a variable: 2 The 3 in the term 3s is the coefficient Expression: one or more terms added or subtracted together. 2 3x, 3x + 2, 4y – 8y + 3 Distributive Prop: a(b + c)= ab + ac A number multiplying parentheses must be multiplied by every term within it. -2(3x – 8) = -6x + 16 2 3 2 -3x(x – 4x + 5) = -3x + 12x - 15x Nominal Words Polynomial – Many Termed Expression Monomial – One Term Expression: 3x Binomial – Two Term Expression: 3x + 5 3 2 Trinomial – Three Terms: -3x + 12x - 15x Substitution Principle: a = b If a = b, then a & b substituted for each other since they are equal & the same value. Evaluate 2(x – 3) if x = 7. 2(7 – 3) = 2(4) = 8 Powers or exponents are a way to show repeated (exponent) multiplication. Use the form: base 3 Example: 2 = 2 * 2 * 2 A negative number raised to a power will be – if the power is odd and + if the power is even, because the –‘s pair off and cancel each other out. 3 3 Example. (-3) = (-3)(-3)(-3) = -3 = -27 Scientific Notation 5.34 x 10 5 A way to write very large or very small numbers in a 1 2 compact way. Note that 12= 1.2x10 & 120= 1.2x10 In Scientific Notation, we move the decimal to the right or left so that a single digit is to the left of decimal. As per normal decimal rules, each time we move a decimal to the right we've multiplied by 10 (1.2 x 10 = 12.0) & each time we move to the left we've divided by 10 (13.4 / 10 = 1.34). Note that a negative power is the same as the reciprocal, so -2 multiplying by 10 is the same as dividing by 100. Two examples: 3 1364 = 1.364 x 10 -2 .0267 = 2.67 x 10 Multiplying numbers in Scientific Notation follows the same principles: 3 4 (1.2 x 10 )(9.0 x 10 ) 3 4 = (1.2 x 9.0 )( 10 x 10 ) 7 = 10.8 x 10 8 = 1.08 x 10 Evaluate 3a  5b  1 given a = 7, and b =2 4a  11b 3  7  5  2  1 21  10  1 32 1   5 4  7  11  2 28  22 6 3 4.3 Equations An equation has two expressions equal to one another: ( 2x = 10 ) An = symbol is like a balance scale or teeter totter with each side of the = having the same weight. In this case 2x lbs weighs the same as 10 lbs. The objective of solving equations is to get the variable all by itself on one side of the =. Given 2x=10, we want to know what x=. To do this we can Add the same thing to both side & it will still be = , just like a teeter tooter. We can also subtract the same thing from both side, Multiply both sides by the same thing, or divide both sides by the same thing. Remember - whatever we do to one side of an equation, we must do to both sides. Use four Properties of Equality to solve equations Addition: We can add the same amount to each side of an equation and it’s still true. x – 4 = 10 +4 +4 x = 14 4.1 Variables We use a lower case letter to represent an unknown, or variable, value. Typically the first letter of the word a variable represents is used or, by default, we use x & y. For example, in the equation d = r*t, d - stands for the distance traveled, r - is the rate of the car (it's speed) t – is the time traveled 4.2 Expressions Term: numbers and/or variables that are multiplied and/or divided together 4, 3x, 5xy , 5z/r t Like terms: Two or more terms that contain the same form of the variable(s). The same variables must be multiplied together and each must be to the same power. 2 2 4 & 90, 4x & 12x, 4xy & 7xy e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc 2 3 -2- Printed 4/8/2009 Algebra A Cheat Sheet The opposite of subtracting 4 is adding 4. Subtraction: We can subtract the same amount from each side of an equation and it’s still true. x + 12 = 16 - 12 -12 x = 4 The opposite of adding 12 is subtracting 12 Multiplication: We can multiply each side of an equation by the same thing and it’s still true. Given    Zero Product: ab = 0 iff a = 0 or b = 0 Definition of Division: a/b = a * (1/b) Definition of Subtraction: a + (-b) = a – b Note that the Definition of Division says that dividing is the same as multiplying by the reciprocal, as above 4.4 MultiStep Equations If there are variables on both sides, you must go through multiple steps to solve. Start with an equation -> distribute parenthesis combine like terms on the same side of = Add / Subtract to get all variables to left of = and all numbers to the right. Divide both sides to ‘cancel’ multiplying. Word Problems 1. Visualize Problem – Draw a picture or diagram. 2. Identify Variables – Read the last sentence. It’s usually a question & the variable(s) are the answer to the question. Be SURE to include units in the definition of the variable. ( s – hours Sally drove) 3. Translate words into math equation(s) – Each sentence is typically one equation – though not always. Every number should be used in the equation(s). 4. Check your Equations  Do the units check out? Write the equation with units next to each variable & number. Units cancel out like fractions, so 30 miles/hour * 2 hours is 60 miles (the hour/hour cancel out). Do units equate? In order to add, subtract, or equate (=) you must have the same units ( $ + $ = $, feet = feet, hours – hours = hours, etc). I can’t add hours to dollars – it makes no sense. Does the equation make sense? Pick easy numbers that make sense in real life & plug them into the equation. Does the answer sound reasonable? y + 3(2y + 6) = 5y + 20 y + 6y + 18 = 5y 7y + 18 = 5y -5y - 18 -5y – 18 2y /2 y = -9 = -18 /2 done x 7 4 x (4)*  7 *(4) 4 x = 28 The opposite of dividing by 4 is multiplying by 4 Division: We can divide each side of an equation by the same thing and it’s still true. Given 6x = 30 ( 1 1 )* 6x = 30*( ) 6 6 x = 5 The opposite of multiplying by 6 is dividing by 6 (same as multiplying by reciprocal 1/6. (6 / 6 = 1). Canceling a Fraction Multiply each side by the reciprocal of x’s coefficient. 2 3 2 3 x = 4 , ( )* x = 4*( ), x = 6 3 2 3 2 Properties of Equality There are other properties of equality that we intuitively know and use while solving equations. The full list of these properties and their formal names are         Reflexive: a=a Symmetric: If a = b then b = a Transitive: If a = b and c = b then a = c Addition: If a = b then a + c = b + c for any c Subtraction: If a = b then a – c = b – c for any c Multiplication: If a = b ac = bc for any c Division: If a = b and c ≠ 0, then a/c = b/c Proportion: If a/b = c/d then ad = bc  Operations & Order A few more formal definitions and properties that may appear on some tests are:   Cancellation of Add: a + c = b + c then a = b Cancellation of Mult: ac = bc so a = b -3- 5. Solve equation(s) and check solutions - Use substitution or elimination if you have two equations. Do the answers sound reasonable? If you plug (substitute) the solutions back into the equation(s), do they work? Word problems Printed 4/8/2009 e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc Algebra A Cheat Sheet The following shows the ‘translation’ of some math symbols into their equivalent words. You should be able to translate words into ‘math’, and math equations back into words. Math is a language just like Spanish, et al. + sum of, more than, greater, rise - minus, difference, less than, lower, * product, times, of, ÷ quotient, divided by, = is, solution to the inequality, so 4 is included within x ≤ 4 because it = 4, whereas it is not included in x < 4, thus the ‘hole’ in the graph with an open circle for the endpoint. Solving inequalities is exactly the same as solving equations EXCEPT – if you multiply or divide both sides by a negative number; you flip the direction of the inequality. Thus given -3x ≤ 9 we get x ≥ -3. Compound inequalities And - both inequalities must be true, so the solution is where the graphs overlap. Given: x < 4 and x ≥ -2, individual graphs are: <------------------------------○ (x < 4 ) ●--------------------------------> (x ≥ -2 ) <--|---|---|---|---|---|---|---O---|---|---|---|---|---|---|---|---|-> Solution: ●-----------------------○ (x < 4 & x ≥ -2 ) An AND graph is typically a segment as above, but it can be a ray or no solution if they don’t overlap. The above can also be written as ( -2 ≤ x < 4 ), which is the two inequalities written into one. It should always be written in the order of the number line, so the lesser number is to the left & the greater number on the right of the variable in the middle. Or – Only one of the inequalities must be true, so the solution is anywhere either of the individual inequalities is true. Given: x > 3 or x ≤ -2, the individual graphs: ○-----> (x > 3 ) <---------● (x ≤ -2 ) <--|---|---|---|---|---|---|---O---|---|---|---|---|---|---|---|---|-> Solution: <---● ○--> (x > 3 or x ≤ -2 ) 4.5 Ratio, Rate, Proportion, Percent, Vari Rate tbd Ratio A ratio simply a way to state the relationship between two values as a fraction. i.e if there are 15 girls and 15 21 boys in a class the ratio of girls to boys is or it 21 can also be written as 15 : 21. Proportion A proportion is sets two ratios equal to one another, so if the ratio of girls to boys in the Freshman class is equal to that in the classroom, and there are 150 girls in the entire class, then 15 150 . There must be  21 210 210 boys in the entire class for this to be true. Cross Multiply Proportions can be solved by 'cross multiplying'. In the above example 15*210 = 21*150. Also: x 2 -> 3x = -10, therefore x = -10/3  5 3 Percent Problems TBD Direct & Indirect Variation tbd 5.1 Two Variable Equations Two variable equations with x & y An equation like y = 2x - 3 shows a relationship between x and y where y is two times whatever x is minus three. If x = 4, then substitute into equation to find y. Two times 4 is 8, subtract 3 so y = 5. We can say y depends upon x. If you know x, you can find what y is. y is the dependent variable (vertical axis on grid) x is the independent variable (horizontal axis). 4.6 Inequalities Four inequality symbols Meaning < Less than > Greater than ≤ less that or equal to ≥ Greater than or equal to Graph <---○ ○----> <---● ●----> 5.2 Tables & Graphs Cartesian Coordinates ( x, y ) Intersection of two real number lines at origin. x-axis is horizontal, y-axis is vertical. ‘x’ measures how far to the right (+) or left (–) of the origin the point is. ‘y’ Printed 4/8/2009 Note that x < 4 means x is less than 4. When graphing, a solid dot indicates that the end point is a e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc -4- Algebra A Cheat Sheet measures up (+) or down (-). An Ordered Pair (x,y) is a coordinate showing a relationship between two variables x & y. Two points (x1,y1) and (x2,y2) define a line. Quadrants: I (+,+), II (-,+), III (-,-), IV (+,-) Quad II y axis Quad I Vertical line test for functions says that if a vertical line crosses two points on the graph of a relation, it is not a function. This means that there a two different output values (y) given a single input value (x), which is not allowed Domain: all valid values for input (the independent variable - commonly x). When finding the domain of a function, we remember that no division by 0 is allowed. If the function is continuous like a line, it’s simply how wide the graph is (lowest to highest value) Range: all valid values for output (dependent variable - commonly y). If the function is continuous, it’s simply how tall the graph is (lowest to highest value) Origin (0,0) x axis 5.4 Slope Intercept Quad III Scatter Plot Scatter plots: pairs of data Choosing scales (units per tick mark) for graphs x y 0 -3 4 5 Graphing a line using a Table of values Given y = 2x - 3, we can easily graph by choosing any two values for x (say 0 and 4) and calculating, using the equation, what y is. Plot these two points on a graph and two points make a line. x 0 4 y -3 5 There are infinite points where y = 2x -3 on this line, but it only takes two to make the graph. Quad IV Slope intercept form: y = mx + b, Where m is slope & b is y-intercept. So for y = 3x – 4, m = 3 & b = -4 The slope is the ‘rate of change’ between x & y in this equation, for every 1 that 1 increases, y increases by 3. Likewise, if x decreases by 1, y decreases by 3. Given m = -2 and b = 6, the equation is y = -2x + 6 Slope: m= rise/run = y ( y 2  y1 )  x ( x2  x1 ) So for line through ( -3, 1 ) and ( 3, 4 ) m= (4  1) 3  = 1 / 2. (3  (3)) 6 5.3 Functions & Relations Relation: any set of ordered pairs. Inputs may have same output Function: rule establishing 1 to 1 relationship between input and output. Function notation: uses f(x) = x + 3x – 6 2 rather than y = x + 3x – 6 So f(x) takes the place of y A function can be expressed by an equation or simply as a list of ordered pairs (which are just points on a graph). The below set is a relation: {(-1,7), (0,3), (1,5), (1,7),(2,8)} because the input x=1 has two different outputs of 5 or 7. If we remove the point (1,7), it is a function. 2 ( + ) slope is uphill left to right ( - ) slope is downhill ( 0 ) slope is horizontal (undefined) slope is vertical because /0 Checking if a line goes through a point Every point on a line is a solution to the equation, so we can substitute the (x,y) for the point into the equation and check if it’s ‘true’. Is the point (-2, 1) on the line y = x + 3? 1 = -2 + 3 1 = 1 so yes Does y = ½ (x + 4) pass through the point (2, 4)? 4 = ½ (2 + 4) 4 ≠ 3 , so no. Equation of a line given point & m Given a point (-3, 4) and slope m = 3, we find the equation of the line by substituting into y = mx + b and solving for our unknown b. (Since the point is on the line, it must be a solution to our equation) e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc -5- Printed 4/8/2009 Algebra A Cheat Sheet y = 3x + b (since m = 3) 4 = 3(-3) + b thru (-3,4) b = 4 + 9 = 13 Then write equation with m & b: y = 3x + 13 When graphing linear inequalities, solve the equation for y and graph the line as though it were an equation. If the inequality is ≤ or ≥ use a solid line as normal because the points on the line are also solutions to the inequality. If the inequality is a < or >, then use dashed lines because the points on the line aren’t actually solutions to the inequality. The solution, or set of points that satisfies the inequality condition is far more than just the line though, it's actually a shaded area of half the coordinate plane. Given a > or ≥ inequality, shade the top of the line because y > anywhere up there. Given a < or ≤ inequality, shade the bottom. Graph < - - - - - line shaded down > - - - - - line shaded up ≤ ______ solid line shaded down ≥ ______ solid line shaded up Equation of a line given two points Given two points, say (-4, 4) and (2, -2), we can write the equation of the line by first finding the slope m = (y2-y1)/(x2-x1) as above, and then substituting m and one of the two points into y = mx + b, as above. m = ( -2 – 4 ) / ( 2 – (-4)) = -6 / 6 = -1 y = mx + b; so (4) = (-1)(-4) + b, b = 0 so y = -x + 0 5.5 ||, _|_ , Horz and Vert Lines Parallel lines have the same slope y = 4x + 3 and y = 4x -10 are || Two horizontal lines (y = 2 & y = 5) are || Two vertical lines (x = -3 and x = 7) are || One of each (y = 2 & x = 7) are _|_ Perpendicular lines form right angles and slopes are negative reciprocal (flip fraction and opposite sign) y = 2x – 3 and y = -1/2x + 4 are _|_ Horizontal lines are always of the form y = a, because y is the same no matter what x is. So the line y = 0 is the x axis & y = 4 is 4 above the x axis. Vertical Lines are of the form x = a & x always has the value no matter what y is. The y axis is the line x=0 because x=0 everywhere along the y axis. The line x = -6 is a vertical line 6 to the left of the y axis. 5.9 Absolute Value |ax + b| = c Since | | yields a + number, c must be≥ 0. If ax+ b ≥0, then the | | does nothing, but if ax + b < 0, the | | gives the positive value, which is = –(ax + b). An absolute value equation like | 2x – 6| = 4 is two equations in one: 2x – 6 = 4 and also -(2x – 6) = 4 x = 5 or x = -2 | | inequalities: |ax + b| < c yields ax + b < c & ax + b > -c ( since / -1) 7.1 Systems by Graphing Systems of Equations are two or more linear equations and a solution to a system is the pair (x, y) that make all equations true. Since a line is simply a collection of all the solution points for an equation (i.e. every point on a line is a solution to the equation), the solution to a system of equation is the point where the lines cross. It’s the only point on both lines. Checking solutions: A point like (3, -2) is the solution to a system of equation only if it is a solution to BOTH equations, so substitute into each & it must make BOTH equations true. y = -x + 1 : (-2) = -(3) + 1 : -2 = -2 is true y + 2x = 4 : (-2) + 2(3) = 4 : -2 + 6 = 4 is true So this point is the solution to the system Special cases for systems Parallel lines (same slope) never intersect so they have no solutions. 5.6 Point Slope Form; y – y1 = m(x – x1) This is another way to get an equation given point and slope. Given a line through (x1, y1) with slope m, we can use form above, if m = .5 through (-2, 2): y – 2 = .5(x - -2) : y – 2 = .5x + 1 & solving for y: y = .5x + 3 5.7 Standard Form: Ax + By = C Also called General form. Just another way to write the equation of a line with both x & y variables on left side of = & the number on the right. We can graph using x & y intercepts since we know at the y intercept x = 0 and at the x intercept y = 0. Substitute these into equation and find the two points to graph. So for: 3x + 2y = 6 , Yint( 0, 3) Xint (2, 0) Plot these two points & draw line. Using x & y intercepts is really a specialized form of using a table of values to plot the line. 5.8 Linear Inequalities e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc -6- Printed 4/8/2009 Algebra A Cheat Sheet Two equations of the same line overlap and have infinite solutions. inequality is not just a line, but a shaded area above (> or ≥) or below (< or ≤ ) the line, the points in common between the two inequalities is the area where the shading overlaps. 7.2 Systems by Substitution Substitution: Given two equations, A & B, solve for y or x in one equation & substitute into the other equation. A: y = x – 3 B: 2x – y = 10 A -> B: 2x – (x – 3) = 10 Since BOTH equations are true at the solution point, and y = x – 3 in A, we substitute x – 3 for the y into equation B. This gives us a new equation with one variable & we can solve for x. Solve this equation for x: 2x – x + 3 = 10 : x = 7 Then substitute x = 7 back into equation A or B to find y at the solution point. Into A: y = 7 – 3, so y = 4 Solution is the point x = 7 & y = 4 or ( 7, 4) 11 tbd Basic Geometry 14 Probability tbd 3.5 Exponents As above, powers or exponents are a way to show (exponent) repeated multiplication in the form: base 3 Example: x = x * x * x Given this basic definition, it’s easy to see why the simple properties below work. For example: 3 4 7 Product of powers: x x = (x*x*x)( x*x*x*x) = x 7.3 Systems by Elimination Arrange or multiply to get opposite coefficients of a variable and then add the two equations to eliminate x or y. A: 2x + 5y = 4 B: -2x + y = -10 A + B: (2x – 2x) + (5y + y) = (4 – 10) 6y = -6 : y = -1 Since y = -1, substitute it back into equation A or B to find x at the solution point, just like above. Into A: 2x + 5(-1) = 4 -> 2x = 4 + 5 : x = 9/2 Solution is (9/2, -1) Elimination by Multiplication Given: A: 2x + 3y = 7 B: 5x + 4y = 14 We note that adding equations will not cancel out either of the variables. We get equal coefficients by multiplying both sides of equations by a number to get equal #’s (kinda like common denominators). A: 5(2x + 3y) = 5(7) -> 10x + 15y = 35 B: -2(5x + 4y) = -2(14) -> -10x - 8y = -28 7y = 7 Then substitute y = 1 back into A or B to find x. x5 x  x  x  x  x 3 =x  xx x2 x2 xx 1 Negative Exp:   3 = x-3 5 xxxxx x x Quotient of powers: Exponent Properties 0 x = 1 anything to the 0 is 1 -a a x = 1/x Negative exponents 1 st x = x anything to the 1 power is itself a b (a+b) x x =x product of powers: add exponents a b (a - b) x /x =x quotient of powers: subtracts exp. a b ab (x ) = x Power of a power: multiply exponents a a a (xy) = x y Everything w/in ( ) is raised to power x x (1/b) (a/b) = b x Definition of a fractional power a 1/b = (x ) = b xa 6.1 Polynomials & Quadratics In Linear Equations, each variable has a power of 1, like: y = 2x – 3. Expressions and equations with powers greater than 1 also have names according their largest power (degree). Name Degree Example Constant 0 6 Linear 1 3x – 5 2 Quadratic 2 4x – 2x + 6 3 Cubic 3 2x – 2x + 6 We order terms in a polynomial in descending order of power, like the above examples and the coefficient of the leading term is the leading coefficient. + & - Polynomials Just collect like terms. 7.4 Systems of Inequalities As with a System of Equations, the solution is the points (x, y) that make both inequalities true (they satisfy both conditions. Since the graph of an e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc -7- Printed 4/8/2009 Algebra A Cheat Sheet (2x – 2x + 6 ) – (3x – 5x + 4) = -x + 3x + 2 x & ÷ Polynomials Distribute as per normal. 3 4 2 -3x(2x – 4x + 6 ) = -6x + 8x -18x When dividing, we use property 3 3 3 Given :x – x – 6, start with open parentheses 2 (x )(x ) the first terms must multiply to x Using FOIL, the (Outside + Inside) will add to middle term (-x) and the Last terms will multiply to -6, so we need two numbers that multiply to -6 and add to -1, which is -3 and 2, therefore 2 Factors of x – x – 6 are (x – 3)(x + 2) Just FOIL factors back together to check if correct. Factoring ax + bx + c With a non-one 'a' coefficient, we may have many combinations of factors that could work and we may need to try many combinations. Understanding FOIL is still the key to understanding factoring. 2 Given :2x + 7x – 15, we see 2 is prime, so (2x )(x ) Factors of -15 are 1(-15), -1(15), 3(-5), or -3(5). Mentally or on paper we must put each pair of factors into either the first or second binomial until the (outside + inside) is 7x. Answer (2x – 3)(x + 5) Quadratic Equations ax + bx + c = 0 Given an equation 2x + 7x – 15 = 0, we know it factors as (2x – 3)(x + 5) = 0 from above. The question is now, what values of x ‘solve’ this equation, or make it true? To find out, we use the zero product property: If ab = 0, then a = 0 or b = 0 So given (2x – 3)(x + 5) = 0 (2x – 3) = 0 or (x + 5) = 0 Solve each & we get solutions x = 3/2, -5 Quadratic Formula : ax + bx + c = 0 A method for finding solutions to general quadratic equations that always works, even with decimal and imaginary solutions. Simply substitute coefficients into formula: Solutions are: 2 2 2 2 2 ab a b   c c c 15n 2  6n 15n 2 6n So    5n 2 3n 3n 3n Multiplying Binomials Every term must be multiplied by every other term. Use FOIL : First, (Outside + Inside), Last So for (2x – 3)(x + 8) we have 2 2x + (16x – 3x) – 24 2 Answer: 2x + 13x – 24 If we have > two terms in polynomial, distribute: 2 So (2x – 1)( 3x + x - 4) 2 2x( 3x + x - 4) Distribute & 2 – 1( 3x + x - 4) Then collect like terms Special Products 2 2 2 (a + b) = a + 2ab + b Perfect Sq. Trinomial 2 2 a - b = (a + b)(a - b) Difference of 2 Squares 3 3 2 2 a + b = (a + b)(a - ab + b ) 3 3 2 2 a - b = (a - b)(a + ab + b ) Factoring – Greatest Common Factor (GCF) The biggest factor that divides into each term in a polynomial. Factor numbers, then get the lowest power of each variable. So for 5h p + 10h p – 15hp 2 5 is biggest #, h is lowest h, & p is lowest p 2 GCF is 5hp To factor the polynomial, divide each term by GCF 3 3 2 3 2 5h 3 p 3 10h 2 p 3 15hp 2 5hp (   ) 5hp 2 5hp 2 5hp 2 2 = 5hp (h p + 2hp – 3) Factoring – By Grouping Just as 2 is a factor common to 2x + 4 = 2(x + 2). The binomial (3x – 7) is common to 2x(3x – 7) + 4(3x – 7) and it can be factored as (3x – 7)(2x + 4). Likewise, the expression x + 3x – ax – 3a 2 3 2 = x x + 3x – ax – 3a Factoring x + bx + c Factoring is the opposite of multiplying, so we need to find two binomials that FOIL (multiply) together to get back to the given quadratic. e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc 2 2 x  b  b 2  4ac 2a 6.3 Rational Exp & Equation tbd 3 2 6.6 Radicals Like exponents, there are a number of properties of radicals that we can use to simplify expressions. 2 4 = 2, 9 = 3 , etc. So the square root is the -8- Printed 4/8/2009 Algebra A Cheat Sheet number that, when squared, gives the original number, so FOIL for two binomials 4 = 2 because 22 =4 (2  5 2 )(1  3 2 ) 2 2  1 (2  3 2  5 2  1)  5 2  3 2 2  (1 2 )  15 2  28  2 Now (-2) = 4 also, so 4 can be + or – 2, but we usually just state positive root unless it’s in an 2 equation like x = 16, where x = +/-4. Properties of Square Roots: Product: ab  a b 3 12  36  6 5 5 5   16 4 16 a  b a b 2 12  2 4 3  4 3 & Quotient: 2 so, Solving ax + c = 0 Since for x = k, x = +/- k , we can solve for the variable and take the square root of each side, 2 2 2 25z – 96 = 4, 25z = 100, z = 4, z = +/-2 Rationalizing the denominator A radical shouldn't be in denominator, so we multiply top & bottom of fraction to get an equivalent fraction. 2 25 3  5 3  5  3 5 3 5 3    3 3 3 32   Solving Square Root Equations Given x = 4, we can readily say x =16, which yields another property of the =. When solving equations we know we can +, -, x, / the same thing to both sides of the =. Likewise, we can square both sides of the equation so Given  x  = 4 & x =16. 5n  7 =2,  5n  7  =2 , 5n – 7 = 4 2 2 2 Distance Between two points For a right triangle with legs of length a & b, the Pythagorean Theorem states that: 2 2 2 : a +b =c where c is the hypotenuse If given two points (3, 2) and (7, 8 ), the distance between them will be the x 2  y 2 (4) 2  (6) 2 = 7.21 a+b (7  3) 2  (8  2) 2 = Add, Subtract, Mult & Divide c With radical expressions we collect like terms and multiply using distribution and FOIL just like we're used to, so: 3 5 2 3  3 5 3 2 3  6  5 3 e285cadb-4cd1-4108-992d-6f5bd06b5e1f.doc   -9- Printed 4/8/2009