Beginning Algebra Summary Sheets
1. Numbers 2
1.1. Number Lines 2
1.2. Interval Notation 3
2. Inequalities 4
2.1. Linear with 1 Variable 4
3. Linear Equations 5
3.1. The Cartesian Plane 5
3.2. Graphing Lines 6
3.3. Intercepts and Slope 7
3.4. Finding the Equation of a Line 8
4. Systems of Linear Equations 9
4.1. Definitions 9
4.2. Solving by Graphing 9
4.3. Solving by Substitution 10
4.4. Solving by Addition or Subtraction 11
5. Word Problems 12
5.1. Solving 12
6. Polynomials 13
6.1. Definitions 13
6.2. Multiplication 14
6.3. Division 15
7. Factoring 16
7.1. GCF (Greatest Common Factor) 16
7.2. 4 Terms 17
7.3. Trinomials: Leading Coefficient of 1 18
7.4. Trinomials: All 19
7.5. Perfect Square Trinomials & Binomials 20
7.6. Steps to Follow 21
8. Quadratics 22
8.1. About 22
8.2. Graphing 22
8.3. Solve by Factoring 23
8.4. Solve with the Quadratic Equation 24
9. Exponents 25
9.1. Computation Rules 25
9.2. Scientific Notation 26
10. Radicals 27
10.1. Definitions 27
10.2. Computation Rules 28
11. Rationals 29
11.1. Simplifying Expressions 29
11.2. Arithmetic Operations 30
11.3. Solving Equations 31
12. Summary 32
12.1. Formulas 32
12.2. Types of Equations 33
12.3. Solve Any 1 Variable Equation 34
Numbers
|Number Lines |
|Number Lines | ( ) – If the point is not included | |
| |[pic]– If the point is included | |
| |– Shade areas where infinite points are included | |
|Real Numbers |Points on a number line |[pic] |
| |Whole numbers, integers, rational and irrational numbers | |
|Positive Infinity |An unimaginably large positive number. (If you keep going to the | |
|(Infinity) |right on a number line, you will never get there) |[pic] |
|Negative Infinity |An unimaginably small negative number. (If you keep going to the | |
| |left on a number line, you will never get there) |[pic] |
|Interval Notation |
|Interval Notation |1st graph the answers on a number line, then write the interval notation by following your shading from left to right |
|(shortcut, instead of |Always written: 1) Left enclosure symbol, 2) smallest number, 3) comma, 4) largest number, 5) right enclosure symbol |
|drawing a number line) |Enclosure symbols |
| |( ) – Does not include the point |
| |[ ] – Includes the point |
| |Infinity can never be reached, so the enclosure symbol which surrounds it is an open parenthesis |
|[pic] |[pic] |[pic] |[pic] |
Inequalities
|Linear with 1 Variable |
|Standard Form |[pic] |[pic]2x + 4 > 10 |
|Solution |A ray |[pic]x > 3 0 1 2 3 |
|Multiplication Property of|When both sides of an inequality are multiplied or divided by a negative number, the|[pic] |
|Inequality |direction of the inequality symbol must be reversed to form an equivalent | |
| |inequality. | |
|Solving |Same as Solving an Equation with 1 Variable (MA090), except when both sides are | [pic] |
| |multiplied or divided by a negative number |-3 -2 -1 |
| |Checking |[pic] |
| |Plug solution(s) into the original equation. Should get a true inequality. | |
| |Plug a number which is not a solution into the original equation. Shouldn’t get a | |
| |true inequality | |
Linear Equations
|The Cartesian Plane |
|Rectangular Coordinate |Two number lines intersecting at the point 0 on each number line. |Quadrant II Quadrant I |
|System |x-axis - The horizontal number line |[pic] |
| |y-axis - The vertical number line |Quadrant III Quadrant IV |
| |origin - The point of intersection of the axes | |
| |Quadrants - Four areas which the rectangular coordinate system is divided into | |
| |Ordered pair - A way of representing every point in the rectangular coordinate | |
| |system (x,y) | |
|Is an Ordered Pair a |Yes, if the equation is a true statement when the variables are replaced by the |Ex x + 2y = 7 |
|Solution? |values of the ordered pair |(1, 3) is a solution because |
| | |1 + 2(3) = 7 |
|Graphing Lines |
|General |Lines which intersect the x-axis contain the variable x | |
| |Lines which intersect the y-axis contain the variable y | |
| |Lines which intersect both axis contain x and y | |
|Graphing by plotting |Solve equation for y |[pic] |
|random points |Pick three easy x-values & compute the corresponding y-values |x |
| |Plot ordered pairs & draw a line through them. (If they don’t line up, you |y |
| |made a mistake) | |
| | |–1 |
| | |4 |
| | | |
| | |0 |
| | |3.5 |
| | | |
| | |1 |
| | |3 |
| | | |
|Graphing linear |Plot the point |[pic][pic] |
|equations by using a |Starting at the plotted point, vertically move the rise of the slope and |Point = 7/2 |
|point and a slope |horizontally move the run of the slope. Plot the resulting point |Slope = –1/2 |
| |Connect both points | |
|Intercepts and Slope |
|x-intercept |where the graph crosses the x-axis |[pic] |
|(x, 0) |Let y = 0 and solve for x | |
|y-intercept |where the graph crosses the y-axis |[pic] |
|(0, y) |Let x = 0 and solve for y | |
|Slope of a Line |The slant of the line. |[pic] |
| |[pic][pic] | |
|Properties of Slope |Positive slope - Line goes up (from left to right). The greater the | |
| |number, the steeper the slope | |
| |negative slope - Line goes down (from left to right). The smaller the | |
| |number (more negative), the steeper the slope. | |
| |Horizontal line - Slope is 0 | |
| |vertical line - Slope is undefined | |
| |parallel lines - Same slope | |
| |perpendicular lines - The slope of one is the negative reciprocal of the | |
| |other | |
| |Ex: m = –½ is perpendicular to m = 2 | |
|Standard Form |ax + by = c |[pic] |
| |x and y are on the same side | |
| |The equations contains no fractions and a is positive | |
|Slope-Intercept Form |y = mx + b, where m is the slope of the line, & b is the y-intercept |[pic] |
| |“y equals form”; “easy to graph form” | |
|Point-Slope Form |y – y1= m(x – x1), where m is the slope of the line & (x1, y1) is a point|[pic] |
| |on the line | |
| |Simplified, it can give you Standard Form or Slope-Intercept Form | |
|Finding the Equation of a Line |
|If you have a horizontal line… |The slope is zero |Ex. y = 3 |
| |y = b, where b is the y-intercept | |
|If you have a vertical line… |The slope is undefined |Ex. x = -3 |
| |x = c, where c is the x-intercept | |
|If you have a slope & |Plug directly into Slope-Intercept Form |[pic] |
|y-intercept… | | |
| | | |
| | | |
|If you have a point & a slope… |Method 1 |[pic] |
| | | |
| |Use Point-Slope Form | |
| |Work equation into Standard Form or Slope-Intercept Form | |
| |Method 2 |[pic] |
| | | |
| |Plug the point into the Slope-Intercept Form and solve for b | |
| | | |
| | | |
| |Use values for m and b in the Slope-Intercept Form | |
|If you have a point & a line that |Determine the slope of the parallel or perpendicular line (e.g.. if |[pic] |
|it is parallel or perpendicular |it is parallel, it has the same slope) | |
|to… |If the slope is undefined or 0, draw a picture | |
| | | |
| |If the slope is a non-zero real number, go to If you have a point & | |
| |a slope… | |
|If you have 2 points… |Use the slope equation to determine the slope |[pic] |
| |Go to If you have a point & a slope… | |
Systems of Linear Equations
|Definitions |
|Type of Intersection |Identical (I) - Same slope & same y-intercept |Identical |
| |No solution (n) - Same slope & different y-intercept, the lines are parallel |Consistent |
| |One point - Different slopes |Dependent |
|Terminology |Consistent System - The lines intersect at a point or are identical. System has at|No solution |
| |least 1 solution |Inconsistent |
| |Inconsistent system - The lines are parallel. System has no solution |Independent |
| |dependent equations - The lines are identical. Infinite solutions | |
| |Independent equations - The lines are different. One solution or no solutions | |
| | |One point |
| | |Consistent |
| | |Independent |
| | | |
|Solving by Graphing |
|1 |Graph both equations on the same Cartesian plane |[pic] |
| |The intersection of the graphs gives the common solution(s). If the graphs | |
| |intersect at a point, the solution is an ordered pair. | |
| | | |
| | |(0,-1) |
| | |[pic] |
|2 |Check the solution in both original equations | |
|Solving by Substitution |
| |[pic] |
| | |
| | |
|Solve either equation for either variable. (pick the equation with the easiest variable | |
|to solve for) | |
|Substitute the answer from step 1 into the other equation | |
| | |
|Solve the equation resulting from step 2 to find the value of one variable * | |
| | |
|Substitute the value form Step 3 in any equation containing both variables to find the | |
|value of the other variable. | |
| | |
|Write the answer as an ordered pair | |
| | |
|Check the solution in both original equations | |
*If all variables disappear & you end up with a true statement (e.g. 5 = 5), then the lines are identical
If all variables disappear & you end up with a false statement (e.g. 5 = 4), then the lines are parallel
|Solving by Addition or Subtraction |
| |[pic] |
| | |
| | |
|Rewrite each equation in standard form | |
|Ax + By = C | |
|If necessary, multiply one or both equations by a number so that the coefficients| |
|of one of the variables are opposites. | |
| | |
| | |
|Add equations (One variable will be eliminated)* | |
|Solve the equation resulting from step 3 to find the value of one variable. | |
| | |
|Substitute the value form Step 4 in any equation containing both variables to | |
|find the value of the other variable. | |
|Write the answer as an ordered pair | |
|Check the solution in both original equations | |
*If all variables disappear & you end up with a true statement (e.g. 5 = 5), then the lines are identical
If all variables disappear & you end up with a false statement (e.g. 5 = 4), then the lines are parallel
Word Problems
|Solving |
| |1 Variable, 1 Equation Method |2 Variables, 2 Equations Method |
|( Understand the problem |In a recent election for mayor 800 people voted. Mr. Smith received three times as many votes as |
|As you use information, cross it out or underline|Mr. Jones. How many votes did each candidate receive? |
|it. | |
|( Define variables |Name what x is (Can only be one thing. When in | |
|Create “Let” statement(s) |doubt, choose the smaller thing) | |
|The variables are usually what the problem is |Define everything else in terms of x | |
|asking you to solve for |Let x = Number of votes Mr. J | |
| |3x = Number of votes Mr. S | |
| | |Let x = Number of votes Mr. S |
| | |y = Number of votes Mr. J |
|( Write the equation(s) | |Usually each sentence is an equation |
|You need as many equations as you have variables | |[pic] |
| |[pic] | |
|( Solve the equation(s) |[pic] |[pic] |
|( answer the question |GO BACK TO YOUR “LET” STATEMENT |Go back to your “Let” statement |
|ANSWER MUST INCLUDE UNITS! |200 = Number of votes Mr. J |200 = Number of votes Mr. J |
| |600 = Number of votes Mr. S |Go back to your “Equations” & solve for remaining|
| | |variable |
| | |[pic] |
| | |600 = Number of votes Mr. S |
| | | |
| | | |
|(check |[pic] |[pic] |
|PLUG ANSWERS INTO EQUATION(S) | | |
Polynomials
|Definitions |
|Term |A constant, a variable, or a product of a constant and one or more variables raised to powers. |
|Polynomial |A sum of terms which contains only whole number exponents and no variable in the denominator |
|Polynomial Name According to Number of |Number of Terms |
|Terms |Polynomial Name |
| |Examples |
| | |
| |1 |
| |Monomial |
| |3x |
| | |
| |2 |
| |Binomial |
| |3x + 3 |
| | |
| |3 |
| |Trinomial |
| |x2 + 2x + 1 |
| | |
|Degree of a Polynomial |Express polynomial in simplified (expanded) form. |
|Determines number of answers |Sum the powers of each variable in the terms. |
|(x-intercepts) |The degree of a polynomial is the highest degree of any of its terms |
|Polynomial Name According to Degree |Degree |
| |Polynomial Name |
| |Examples |
| | |
| |1 |
| |Linear |
| |3x |
| | |
| |2 |
| |Quadratic |
| |3x2 |
| | |
| |3 |
| |Cubic |
| |3x3 |
| | |
| |4 |
| |Quartic |
| |3x4 3x3y |
| | |
|Multiplication |
|Multiply each term of the first polynomial by each term of the second polynomial, and then combine like terms |
|Horizontal Method |[pic] |
|Can be used for any size polynomials | |
|Vertical Method |[pic] |
|Can be used for any size polynomials. | |
|Similar to multiplying two numbers | |
|together | |
|FOIL Method | |
|May only be used when multiplying two | |
|binomials. First terms, Outer terms, | |
|Inner terms, Last terms |[pic] |
|Division |
|Dividing a Polynomial by a Monomial |
|Write Each Numerator Term over the |[pic] |
|Denominator Method | |
|[pic] | |
|Factor Numerator and Cancel Method |[pic] |
Factoring
|GCF (Greatest Common Factor) |
|Factoring |Writing an expression as a product |[pic] |
| |Numbers can be written as a product of primes. Polynomials can be |[pic][pic] |
| |written as a product of prime polynomials | |
| |Useful to simplify rational expressions and to solve equations | |
| |The opposite of multiplying | |
|GCF of a List of Integers |Write each number as a product of prime numbers |[pic]Find the GCF of 18 & 30 |
| |Identify the common prime factors |[pic] |
| |The product of all common prime factors found in Step 2 is the GCF. | |
| |If there are no common prime factors, the GCF is 1 | |
|GCF of a List of Variables |The variables raised to the smallest power in the list |[pic]Find the GCF of x & x2 |
| | |GCF = x |
|GCF of a List of Terms |The product of the GCF of the numerical coefficients and the GCF of |[pic]Find the GCF of 18x& 30x2 |
| |the variable factors |GCF = 6x |
|Factor by taking out the GCF |Find the GCF of all terms |[pic] |
| |Write the polynomial as a product by factoring out the GCF | |
| |Apply the distributive property | |
| |Check by multiplying | |
|4 Terms |
|a + b + c + d = (? + ?)(? + ?) |
|factor by grouping |[pic]Factor 10ax–6xy–9y+15a |
|ARRANGE TERMS SO THE 1ST 2 TERMS HAVE A COMMON FACTOR AND THE LAST 2 HAVE A COMMON FACTOR |10ax + 15a – 6xy – 9y |
|For each pair of terms, factor out the pair’s GCF | |
|If there is now a common binomial factor, factor it out |5a(2x + 3) – 3y(2x + 3) |
|If there is no common binomial factor, begin again, rearranging the terms differently. If no| |
|rearrangement leads to a common binomial factor, the polynomial cannot be factored. |(2x + 3)(5a – 3y) |
|Trinomials: Leading Coefficient of 1 |
|x2 + bx + c = (x + ?)(x + ?) |
|trial & error |[pic] |
|PRODUCT IS C | |
|(x + one number)(x + other number) | |
|Sum is b | |
|Place x as the first term in each binomial, then determine whether addition or subtraction | |
|should follow the variable | |
|[pic] | |
|Find all possible pairs of integers whose product is c | |
|For each pair, test whether the sum is b | |
|Check with FOIL [pic] | |
|Trinomials: All |
|ax2 + bx + c = (?x + ?)(?x + ?) |
|Method 1 (trial & error) |Ex: Factor: [pic] |
|TRY VARIOUS COMBINATIONS OF FACTORS OF AX2 AND C UNTIL A MIDDLE TERM OF BX IS OBTAINED |Product is [pic] Product is -5 |
|WHEN CHECKING. | |
| |[pic] |
|Check with FOIL [pic] | |
| |15x – x = 14x (correct middle term) |
|Method 2 (ac, factor by grouping) |Ex: Factor: [pic] |
|IDENTIFY A, B, AND C |a = 3 |
|Find 2 “magic numbers” whose product is ac and whose sum is b. Factor trees can be very|b = 14 |
|useful if you are having trouble finding the magic numbers (See MA090) |c = –5 |
| |ac = (3)[pic]( –5) = –15 |
| |b = 14 |
|Rewrite bx, using the “magic numbers” found in Step 2 |(15)[pic]( –1) = –15 [pic] |
|Factor by grouping |(15) + (–1) = 14 [pic] |
|Check with FOIL [pic] |“magic numbers” 15, –1 |
| |3x2 + 15x – x – 5 |
| |3x(x + 5) – 1(x + 5) |
| |(x + 5)(3x – 1) |
|Method 3 (quadratic formula) |Ex: Factor: [pic] |
|USE THE QUADRATIC FORMULA TO FIND THE X VALUES (OR ROOTS) |[pic] |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
|For each answer in step 1., rewrite the equation so that it is equal to zero | |
| | |
| | |
| | |
|Multiply the two expressions from step 2, and that is the expression in factored form. | |
| | |
|Check with FOIL [pic] | |
|Perfect Square Trinomials & Binomials |
|Perfect Square |Factors into perfect squares (a binomial squared) |[pic] |
|Trinomials |[pic] | |
| | | |
|a2[pic]2ab + b2 | | |
|Difference of Squares |Factors into the sum & difference of two terms |[pic] |
|a2 – b2 |[pic] | |
|Sum of Squares |Does not factor |[pic] |
|a2 + b2 |[pic] | |
|Difference of Cubes a3 – b3 (MA103) |[pic] |[pic] |
|Sum of Cubes |[pic] |[pic] |
|a3 + b3 (MA103) | | |
|Prime Polynomials (P) |Can not be factored |[pic] |
|Steps to Follow |
|Put variable terms in descending order of degree with the constant term last. |[pic] |
|Factor out the GCF |[pic] |
|Factor what remains inside of parenthesis |[pic] |
|2 terms – see if one of the following can be applied | |
|Difference of Squares | |
|Sum of Cubes | |
|Difference of Cubes | |
|3 terms – try one of the following | |
|Perfect Square Trinomial | |
|Factor Trinomials: Leading Coefficient of 1 | |
|Factoring Any Trinomial | |
|4 terms – try Factor by Grouping | |
|If both steps 2 & 3 produced no results, the polynomial is prime. You’re done ( (Skip steps 5 & 6)| |
|See if any factors can be factored further |[pic] |
|Check by multiplying |[pic] |
Quadratics
|About |
|Standard Form |ax2 + bx + c = 0 |[pic]x2 – 3x + 2 = 0 |
|Solutions |Has n solutions, where n is the highest exponent |[pic] x3 – 3x2 + 2x = 0 (has 3 solutions) |
|Graphing |
|Standard Form |y = ax2 + bx + c |[pic]y = x2 – 9x + 20 |
| |a, b, and c are real constants | |
|Solution |A parabola | |
|Simple Form |y = ax2 |[pic]y = –4x2 |
| |Vertex (high/low point) is (0,0) | |
| |Line of symmetry is x = 0 | |
| |The parabola opens up if a > 0, down if a < 0 | |
|Graph |Plot y value at vertex |[pic]y = –4x2 |
| |Plot y value one unit to the left of the vertex |x |
| |Plot y value one unit to the right of the vertex |y |
| | | |
| | |0 |
| | |0 |
| | | |
| | |–1 |
| | |–4 |
| | | |
| | |1 |
| | |–4 |
| | | |
|Solve by Factoring |
|Zero Factor Property |If a product is 0, then a factor is 0 |[pic]xy = 0 (either x or y must be zero) |
|Solve by Factoring |Write the equation in standard form (equal 0) |[pic]x2 – 3x + 2 = 0 |
| |Factor |1. x(x – 1) (x – 2) = 0 |
| |Set each factor containing a variable equal to zero |2. x = 0, x – 1 = 0, x – 2 = 0 |
| |Solve the resulting equations |3. x = 0, 1, 2 |
|Solve with the Quadratic Equation |
|To solve a quadratic equation that is difficult or impossible to factor |
|[pic] |Ex Radicand is a perfect square |
| |[pic] |
| |Ex Radicand breaks into “perfect square” and “leftovers” |
| |[pic] |
| |Ex Radicand is just “leftovers” |
| |[pic] |
Exponents
|Computation Rules |
|Exponential notation | base xa exponent |[pic] |
|Shorthand for repeated multiplication | | |
|Multiplying common bases |[pic] |[pic] |
|Add powers | | |
|Dividing common bases |[pic] |[pic] |
|Subtract powers | | |
|Raising a product to a power |[pic] |[pic] |
|Raise each factor to the power | | |
|Raising a quotient to a power |[pic] |[pic] |
|Raise the dividend and divisor to the power | | |
|Raising a power to a power |[pic] |[pic] |
|Multiply powers | | |
|Raising to the zero power |[pic] |[pic] |
|One | | |
|Raising to a negative power |[pic] |[pic] |
|Reciprocal of positive power | | |
|When simplifying, eliminate negative powers | | |
|Scientific Notation |
|Scientific Notation |Shorthand for writing very small and large numbers |[pic] |
| |[pic] | |
|Standard Form |Long way of writing numbers |[pic] |
|Standard Form |Move the decimal point in the original number to the left or right so that there is |510. |[pic] |
|to Scientific Notation|one digit before the decimal point |[pic] | |
| |Count the number of decimal places the decimal point is moved in STEP 1 | | |
| |If the original number is 10 or greater, the count is positive | | |
| |If the original number is less than 1, the count is negative | | |
| |Multiply the new number from STEP 1 by 10 raised to an exponent equal to the count |+2 | |
| |found in STEP 2 | |–2 |
| | | | |
| | | |[pic] |
| | |[pic] | |
|Scientific Notation |Multiply numbers together |[pic] |[pic] |
|to Standard Form | | | |
Radicals
|Definitions |
|Roots |Undoes raising to powers |[pic] |
| |[pic] | |
| | | |
| |index | |
| |[pic] radical | |
| |radicand | |
|Computation |If n is an even positive integer, then [pic] |[pic] |
| |The radical [pic]represents only the non-negative square root | |
| |of a. The | |
| |–[pic]represents the negative square root of a. | |
| | | |
| |If n is an odd postivie integer, then [pic] | |
|Notation: |The root of a number can be expressed with a radical or a |[pic]Note, it’s usually easier to compute the root |
|Radical vs. Rational Exponent |rational exponent |before the power |
| |Rational exponents | |
| |The numerator indicates the power to which the base is raised. | |
| | | |
| |The denominator indicates the index of the radical | |
|Computation Rules |
|Operations |Roots are powers with fractional exponents, thus power rules |[pic] |
| |apply. | |
|Product Rule |[pic] |[pic] |
|Quotient Rule |[pic] |[pic] |
|Simplifying Expressions |Separate radicand into “perfect squares” and “leftovers” |[pic] |
| |Compute “perfect squares” | |
| |“Leftovers” stay inside the radical so the answer will be | |
| |exact, not rounded | |
Rationals
|Simplifying Expressions |
|Rational Numbers |Can be expressed as quotient of integers (fraction) where the |[pic]0 = 0/1 |
| |denominator [pic] 0 | |
| |All integers are rational |[pic]4 = 4/1 |
| |All “terminating” decimals are rational |[pic]4.25 = 17/4 |
|Irrational Numbers |Cannot be expressed as a quotient of integers. Is a non-terminating |[pic] |
| |decimal | |
|Rational Expression |An expression that can be written in the form[pic], where P and Q are |[pic], Find real numbers for which this expression |
| |polynomials |is undefined: x + 6 = 0; x = [pic]6 |
| |Denominator [pic]0 | |
|Simplifying Rational |Completely factor the numerator and denominator |[pic] |
|Expressions |Cancel factors which appear in both the numerator and denominator | |
|(factor) | | |
|Arithmetic Operations |
|Multiplying/ |If it’s a division problem, change it to a multiplication problem |[pic] |
|Dividing Rational |Factor & simplify | |
|Expressions |Multiply numerators and multiply denominators | |
|(multiply across) |Write the product in simplest form | |
|Adding/ Subtracting Rational|Factor & simplify each term |[pic] |
|Expressions |Find the LCD | |
|(get common denominator) |The LCD is the product of all unique factors, each raised to a power | |
| |equal to the greatest number of times that it appears in any one | |
| |factored denominator | |
| |Rewrite each rational expression as an equivalent expression whose | |
| |denominator is the LCD | |
| |Add or subtract numerators and place the sum or difference over the | |
| |common denominator | |
| |Write the result in simplest form | |
|Solving Equations |
|Solving by Eliminating the | |[pic] |
|Denominator | | |
| |Factor & simplify each term | |
| |Multiply both sides (all terms) by the LCD | |
| | | |
| | | |
| |Remove any grouping symbols | |
| | | |
| |Solve | |
| |Check answer in original equation. If it makes | |
| |any of the denominators equal to 0 (undefined), | |
| |it is not a solution | |
|Solving Proportions with the|If your rational equation is a proportion, it’s |[pic] |
|Cross Product |easier to use this shortcut | |
|[pic] |Set the product of the diagonals equal to each | |
| |other | |
| |Solve | |
| |Check | |
Summary
|Formulas |
|Geometric | |sum of angles: Angle 1 + Angle 2 + Angle 3 = 180 o |
| |Triangle 1 | |
| |2 3 | |
| |Right Triangle |Pythagorean theorem: a 2 + b 2 = c 2 |
| |c b|(a = leg, b = leg, c = hypotenuse) |
| |a |~The hypotenuse is the side opposite the right angle. It is always the longest side. |
|Other |Distance |distance: d = rt |
| | |(r = rate, t = time) |
|Types of Equations |
| |1 Variable |2 Variables |
|Linear Equations |x – 2 = 0 MA090 |y = x – 2 page 8 |
| |Solution: 1 Point |Solution: Line |
| |0 2 |2 |
|Linear Inequalities |x – 2 < 0 page 4 |y > x – 2 |
| |Solution: Ray |Solution: ½ plane |
| |0 2 | |
|Systems of Linear Equations |[pic] page 10 |[pic] page 10 |
| |Solution: 1 point, infinite points or no points |Solution: 1 point, infinite points or no points |
| | | |
|Quadratic Equations |x2 +5x + 6 = 0 page 34 |y = 2x2 page 22 |
| |Solution: Usually 2 points |Solution: Parabola |
| |-3 -2 0 | |
|Higher Degree Polynomial Equations |x3 + 5x2 + 6x = 0 page 34 |y = x3 + 5x2 + 6x |
|(cubic, quartic, etc.) |Solution: Usually x points, where x is the highest |Solution: Curve |
| |exponent | |
| |-3 -2 0 | |
|Rational Equations |[pic] page 31 |[pic] |
| |Solution: Sometimes simplifies to a linear or |Solution: Sometimes simplifies to a linear or |
| |quadratic equation |quadratic equation |
* To determine the equation type, simplify the equation. Occasionally all variables “cancel out”.
▪ If the resulting equation is true (e.g. 5 = 5), then all real numbers are solutions.
▪ If the resulting equation is false (e.g. 5 = 4), then there are no solutions.
1 Solve Any 1 Variable Equation
-----------------------
m = 0
m = –2
m = –1/2
m1 = 2
m2 = 2 222
m = 1/2
m = undefined
Is it really an equation?
It’s an expression, you can’t solve it. You can factor, expand & simplify it
No
Yes
Make an equivalent, simpler equation
← If the equation contains fractions, eliminate the fractions (multiplying both sides by the LCD)
← If there is a common factor in each term, divide both sides of the equation by the common factor
Solve by “undoing” the equation
← Linear equations can by undone with the addition, subtraction, multiplication & division equality properties
← Quadratics, of the form (x + a)2 = b, can be undone with the square root property
Can the variable be isolated?
Yes
No
Write the equation in standard form
← Make one side equal to zero
← Put variable terms in descending order of degree with the constant term last
Not covered in this class
Can it easily be put in factored form?
Yes
Solve by Factoring
No
Is it a quadratic equation?
Solve with the Quadratic Equation -or-
Solve by Completing the Square
Yes
No
Check solutions in the original equation
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Page 10 of 34
5/7/2013
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