Math Factoids
(thanks to Hoxie H.S., Hoxie, Kansas)
Just about everything you need ... in one spot (are we missing something? let us know)
Number Systems
Scientific Notation 0(zero) 1(one) 2(two) 3(three) 4(four) 5(five) 6(six) 7(seven) 8(eight) 9(nine) 10^1(ten) 10^2(hundred) 10^3(thousand) name American-French English-German million 10^6 10^6 billion 10^9 10^12 trillion 10^12 10^18 quadrillion 10^15 10^24 quintillion 10^18 10^30 sextillion 10^21 10^36 septillion 10^24 10^42 octillion 10^27 10^48 nonillion 10^30 10^54 decillion 10^33 10^60 undecillion 10^36 10^66 duodecillion 10^39 10^72 tredecillion 10^42 10^78 quatuordecillion 10^45 10^84 quindecillion 10^48 10^90 sexdecillion 10^51 10^96 septendecillion 10^54 10^102 octodecillion 10^57 10^108 novemdecillion 10^60 10^114 vigintillion 10^63 10^120 ---------------------------------------- googol 10^100 googolplex 10^googol = 10^(10^100) ----------------------------------------
SI Prefixes
|
|
Roman Numerals
| I=1 | V=5 | X=10 | L=50 | C=100 | D=500 | M=1 000 |
| _ V=5 000 |
_ X=10 000 |
_ L=50 000 |
_ C = 100 000 |
_ D=500 000 |
_ M=1 000 000 |
Examples:
1 = I 2 = II 3 = III 4 = IV 5 = V 6 = VI 7 = VII 8 = VIII 9 = IX 10 = X |
11 = XI 12 = XII 13 = XIII 14 = XIV 15 = XV 16 = XVI 17 = XVII 18 = XVIII 19 = XIX 20 = XX 21 = XXI |
25 = XXV 30 = XXX 40 = XL 49 = XLIX 50 = L 51 = LI 60 = LX 70 = LXX 80 = LXXX 90 = XC 99 = XCIX |
Number Base Systems
decimal binary ternary oct hex
0 0 0 0 0
1 1 1 1 1
2 10 2 2 2
3 11 10 3 3
4 100 11 4 4
5 101 12 5 5
6 110 20 6 6
7 111 21 7 7
8 1000 22 10 8
9 1001 100 11 9
10 1010 101 12 A
11 1011 102 13 B
12 1100 110 14 C
13 1101 111 15 D
14 1110 112 16 E
15 1111 120 17 F
16 10000 121 20 10
17 10001 122 21 11
18 10010 200 22 12
19 10011 201 23 13
20 10100 202 24 14
Root Table (2nd-7th)
| Number | 2nd Power |
3rd Power |
4th Power |
5th Power |
6th Power |
7th Power |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
3 |
9 |
27 |
81 |
243 |
729 |
2,187 |
4 |
16 |
64 |
256 |
1,024 |
4,096 |
16,384 |
5 |
25 |
125 |
625 |
3,125 |
15,625 |
78,125 |
6 |
36 |
216 |
1,296 |
7,776 |
46,656 |
279,936 |
7 |
49 |
343 |
2,401 |
16,807 |
117,649 |
823,543 |
8 |
64 |
512 |
4,096 |
32,768 |
262,144 |
2,097,152 |
9 |
81 |
729 |
6,561 |
59,049 |
531,441 |
4,782,969 |
10 |
100 |
1,000 |
10,000 |
100,000 |
1,000,000 |
10,000,000 |
11 |
121 |
1,331 |
14,641 |
161,051 |
1,771,561 |
19,487,171 |
12 |
144 |
1,728 |
20,736 |
248,832 |
2,985,984 |
35,831,808 |
Fracton to Decimal Conversion Tables
Important Note: any span of numbers that is underlined signifies that those numbers (digits) that are repeated. For example, 0.09 signifies 0.090909... etc.
Only fractions in lowest terms are
listed. For instance, to find 2/8,
first simplify it to 1/4 then search for it in the table below.
| fraction = decimal | |||
| 1/1 = 1 | |||
| 1/2 = 0.5 | |||
| 1/3 = 0.3 | 2/3 = 0.6 | ||
| 1/4 = 0.25 | 3/4 = 0.75 | ||
| 1/5 = 0.2 | 2/5 = 0.4 | 3/5 = 0.6 | 4/5 = 0.8 |
| 1/6 = 0.16 | 5/6 = 0.83 | ||
| 1/7 = 0.142857 | 2/7 = 0.285714 | 3/7 = 0.428571 | 4/7 = 0.571428 |
| 5/7 = 0.714285 | 6/7 = 0.857142 | ||
| 1/8 = 0.125 | 3/8 = 0.375 | 5/8 = 0.625 | 7/8 = 0.875 |
| 1/9 = 0.1 | 2/9 = 0.2 | 4/9 = 0.4 | 5/9 = 0.5 |
| 7/9 = 0.7 | 8/9 = 0.8 | ||
| 1/10 = 0.1 | 3/10 = 0.3 | 7/10 = 0.7 | 9/10 = 0.9 |
| 1/11 = 0.09 | 2/11 = 0.18 | 3/11 = 0.27 | 4/11 = 0.36 |
| 5/11 = 0.45 | 6/11 = 0.54 | 7/11 = 0.63 | |
| 8/11 = 0.72 | 9/11 = 0.81 | 10/11 = 0.90 | |
| 1/12 = 0.083 | 5/12 = 0.416 | 7/12 = 0.583 | 11/12 = 0.916 |
| 1/16 = 0.0625 | 3/16 = 0.1875 | 5/16 = 0.3125 | 7/16 = 0.4375 |
| 11/16 = 0.6875 | 13/16 = 0.8125 | 15/16 = 0.9375 | |
| 1/32 = 0.03125 | 3/32 = 0.09375 | 5/32 = 0.15625 | 7/32 = 0.21875 |
| 9/32 = 0.28125 | 11/32 = 0.34375 | 13/32 = 0.40625 | |
| 15/32 = 0.46875 | 17/32 = 0.53125 | 19/32 = 0.59375 | |
| 21/32 = 0.65625 | 23/32 = 0.71875 | 25/32 = 0.78125 | |
| 27/32 = 0.84375 | 29/32 = 0.90625 | 31/32 = 0.96875 |
Need to convert a repeating decimal to a fraction? Follow these examples:
Note the following pattern for repeating decimals:
0.22222222... = 2/9
0.54545454... = 54/99
0.298298298... = 298/999
Division by 9's causes the repeating pattern.
Note the pattern if zeros preceed the repeating decimal:
0.022222222... = 2/90
0.00054545454... = 54/99000
0.00298298298... = 298/99900
Adding zero's to the denominator adds zero's before the repeating decimal.
To convert a decimal that begins with a non-repeating part, such as
0.21456456456456456..., to a fraction, write it as the sum of the
non-repeating part and the repeating part.
0.21 + 0.00456456456456456...
Next, convert each of these decimals to fractions. The first decimal has
a divisor of power ten. The second decimal (which repeats) is convirted according
to the pattern given above.
21/100 + 456/99900
Now add these fraction by expressing both with a common divisor
20979/99900 + 456/99900
and add.
21435/99900
Finally simplify it to lowest terms
1429/6660
and check on your calculator or with long division.
= 0.2145645645...
Common Units of Measurement and Conversion
Linear Measure
Linear measure is used in determining distances and lengths, widths or thicknesses.
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
5 & 1/2 yards, or 16 & 1/2 feet = 1 rod (rd.)
40 rods = 1 furlong (fur.)
8 furlongs, or 320 rods = 1 mile (mi.)
Avoirdupois Weight
Avoirdupois weight is used in weighing heavy, course articles.
16 drams = 1 ounce (oz.)
16 ounces - 1 pound (lb.)
100 pounds = 1 hundredweight (cwt.)
20 hundredweights = 1 ton (T.)
2,000 pounds = 1 ton
2,240 pounds = 1 long, or gross, ton
7,000 grains (gr.) = 1 pound avoirdupois
Troy Weight
Troy weight is used in weighing precious minerals.
24 grains =1 pennyweight (pwt.)
20 pennyweights = 1 ounce
12 ounces = 1 pound
240 pennyweights = 1 pound
5,760 grains = 1 pound troy
Liquid Measure
Liquid measure is used in measuring the liquid capacity of containers. Also see "Cooking Measures" below.
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
31 & 1/2 gallons = 1 barrel (bbl.)
63 gallons = 1 hogshead (hhd.)
2 barrels = 1 hogshead
7 & 1/2 gallons = 1 cubic foot
Dry Measure
Dry measure is used in measuring the volume of the contents of containers of solids. Also see "Cooking Measures" below.
2 pints = 1 quart (qt.)
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
2 & 3/4 bushels = 1 barrel
Cooking Measures
Cooking measures are a combination of liquid and dry measurements ... rather obviously ... used in the kitchen & when cooking.
a pinch = 1/8 teaspoon or less
3 teaspoons (tsp.) = 1 tablespoon (Tbls.)
6 teaspoons = 2 tablespoons = 1 ounce
4 tablespoons - 1/4 cup (c.) = 2 ounces
8 tablespoons = 1/2 cup = 4 ounces
12 tablespoons = 3/4 cup = 6 ounces
16 tablespoons = 1 cup = 8 ounces
2 cups = 1 pint = 16 ounces (oz.)
4 cups = 1 quart = 32 ounces
4 quarts = 1 gallon (gal.) = 128 ounces
8 quarts = 2 gallons = 1 peck = 256 ounces
4 pecks = 1 bushel
16 ounces = 1 pound
Other Conversion Factors
1 Acre - 0.0015625 square mile; 4.3560 x 10 to the fourth square feet; 0.40468564 hectare
1 Bushel - (U.S.) - 1.244456 cubic feet; 2150.42 cubic inches; 0.035239 cubic meter; 35.23808 liters
1 Centimeter - 0.0328084 foot; 0.393701 inch
1 Circular Mil - 7.853982 x 10 to the negative seventh square inches; 5.067075 x 10 to the negative sixth square centimeters
1 Cubic Centimeter - 0.061024 cubic inch; 0.270512 dram (U.S. fluid); 16.230664 minims (U.S.); 0.999972 milliliter
1 Cubic Foot - 0.803564 bushel (U.S.); 7.480520 gallons (U.S. liquid); 0.028317 cubic meter; 28.31605 liters
1 Cubic Inch - 16.387064 cubic centimeters
1 Cubic Meter - 35.314667 cubic feet; 264.17205 gallons (U.S. liquid)
1 Foot - 0.3048 meter
1 Gallon (U.S. liquid) - 0.1336816 cubic foot; 0.832675 gallon (British); 231 cubic inches; 0.0037854 cubic meter; 3.785306 liters
1 Grain - 0.06479891 gram
1 Gram - 0.00220462 pound (avoirdupois); 0.035274 ounce (avoirdupois); 15.432358 grains
1 Hectare- 2.471054 acres; 1.07639 x 10 to the fifth square feet
1 Inch - 2.54 centimeters
1 Kilogram - 2.204623 pounds (avoirdupois)
1 Kilometer - 0.621371 mile (statute)
1 Liter - 0.264179 gallon (U.S. liquid);0.0353157 cubic foot; 1.056718 quarts (U.S. liquid)
1 Meter - 1.093613 yards; 3.280840 feet; 39.37008 inches
1 Mile (statute) - 1.609344 kilometers
1 Ounce (U.S. fluid) - 1.804688 cubic inches; 29.573730 cubic centimeters
1 Ounce (avoirdupois) - 28.349523 grams
1 Ounce (apothecary or troy) - 31.103486 grams
1 Pint (U.S. liquid) - 0.473163 liter; 473.17647 cubic centimeters
1 Pound (avoirdupois) - 0.453592 kilogram; 453.59237 grams
1 Pound (apothecary or troy) - 0.3732417 kilogram, 373.24172 grams
1 Quart (U.S. dry) - 1.10119 liters
1 Quart (liquid) - 0.946326 liter
1 Radian - 57.295779 degrees
1 Rod - 5.0292 meters
1 Square Centimeter - 0.155000 square inch
1 Square Foot - 0.09290304 square meter
1 Square Inch - 645.16 square millimeters
1 Square Meter - 10.763910 square feet
1 Square Yard - 0.836127 square meter
1 Ton (short) - 907.18474 kilograms
1 Yard - 0.9144 meter
Algebra: Order of Operation
1. Parenthesis, Symbols of Inclusion 2. Upper level math expressions (powers, roots, sine, cosine, tangent, log, etc.) 3. Multiplication / Division (from left to right) 4. Addition / Subtraction (from left to right)
Algebra: Solving First Degree Equations
First Degree Equation: An equation characterized by having a single variable ... which may be repeated within the equation more than once ... which is raised to the first power. Note: plain variable "x" is the same as "x1" ... plain variable "x" is the same as "x" raised to the first power. STEP 1: Eliminate parenthesis (if any) by multiplying through using the distributive property. STEP 2: Eliminate fractions (if any) by finding the common denominator and multiplying the entire equation through by the common denominator. In this example, "12" is the common denominator. NOTE: Notice that the negative sign in front of the compound fraction "(5x - 3)/4" must be distributed through the entire numerator. You are subtracting the total fraction. STEP 3: Combine like terms (if any) on both the left and right sides of the equation. STEP 4: Isolate "x" (or the variable being solved for) to one side of the equation ... and at the same time get non-"x" terms to the opposite side. In this example, you'd subtract "10x" from both sides of the equation (add "-10x" to both sides) ... and subtract "7" from both sides of the equation. STEP 5: Get "x" by its lonesome (get the variable all by itself). In this example, divide both sides of the equation by "5" (or, equivalent, multiply both sides of the equation by the fraction "1/5". STEP 6 (optional): Once an answer is produced, it should be able to be substituted into the original equation ... and make it true. This is a method that can be used to "check" an answer.
Algebra: the Simultaneous Solution of a System of Equations
When an algebraic problem presents itself in which there are two or more equations involving two or more variables ... the multiple equations are refered to as a system. When it is the desire to find a combination of variable values that satisfy each of the equations (make the equations true at the same time), it is referenced as "solving a simultaneous system." If one were to graph the equations ... you'd really be asking the question: "Where do the equations intersect and meet?", "What are the points that are common to both equations simultaneously?" The simplest simultaneous system case involves solving two equations in two unknowns. For example, consider the following system of two equations: 2x + 3y = 12 5x + 2y = 19 There are two common methods used to solve a system. The first method involves combining the equations (using addition or subtraction) in such a way to cause a variable to be eliminated. (Other method shown below.) STEP 1: Examine the two equations to be solved and pick one of the variables to be eliminated. In our example, let's say we pick the "y" variable. STEP 2: Transform the equations (usually using multiplication or division) ... in such a way that the numeric coefficient in front of the "y" variable is the same (absolute) value in both. In this example, you might multiply the top equation by "2" and the bottom equation by "3": 4x + 6y = 24 15x + 6y = 57 Notice that this multiplication has generated a numeric value of "6" as the coefficient of the "y" term in both the top and bottom equation. STEP 3: Combine the two equations (by adding or subtracting them) to eliminate the variable selected in step 1. In this example, subtract the (entire) bottom equation from the top equation ... yielding: -11x = -33 STEP 4: Solve the new equation for the variable remaining. In this instance, divide both sides of the equation by "-11": x = 3 STEP 5: Substitute the value found (in step 4 above) into one of the original equations and find the value of the other variable. For example, substituting "x=3" into the original top equation yields: 2 * ( 3 ) + 3y = 12 6 + 3y = 12 3y = 6 y = 2 STEP 6: The net final solution is the ( x , y ) ordered pair: ( 3 , 2 ). This "x-y" pair will make both the top equation true and, at the same time, the bottom equation true.
Algebra: Solving a Simultaneous System by Substitution
Another common method of solution for a simultaneous system is to utilize substitution. Say, for instance, you want to solve the following system: y = x2 + 2x - 3 2x + 3y = 19 STEP 1: Pick one equation and solve it for one of the two variables. In this example, the top equation is already solved for the variable "y" ... so, without any actual work, it would be the logical selection. In many problems, one would actually have to do some work to solve one of the equations for one of the variables. STEP 2: Replace the variable in the OTHER equation by what it is equal to in step 1. In this example, replace the "y" variable in the bottom equation by "x2 + 2x - 3": 2x + 3 * ( x2 + 2x - 3 ) = 19 STEP 3: Solve the resulting equation: 2x + 3x2 + 6x - 9 = 19 3x2 + 8x - 28 = 0 ( 3x + 14) ( x - 2 ) = 0 x = -14/3 -or- x = 2 STEP 4: The "x" values produced (in step 3 above) must now be substituted into either one of the original equations ... to find the matched-up, corresponding "y" values: when x = -14/3, the matched-up "y"-value is: y = 85/9 when x = 2, the matched-up "y"-value is: y = 5 STEP 5: This particular system of equations has two simultaneous solutions: the "x-y" ordered pair ( -14/3 , 85/9 ) and the "x-y" ordered pair ( 2 , 5 ). Either of these can be substituted into either of the original equations and will produce true statements. A few other notes: Other 1: When solving a system, it is possible that there will be "no" solution. In other words, there will be no "x-y" ordered pair that will make both equations true simultaneously. Other 2: When solving a system, it is a remote possibility that the equations beings solved will share infinitely many "x-y" ordered pairs ... and that all "x-y" pairs that solve one of the equations will also solve the other equation. Other 3: When solving three or more equations simultaneously, it is normally required that one variable (unknown) be eliminated ... followed by a second ... followed by a third ... etc. In other words, it is unlikely that in a single step all variables can be eliminated immediately.
Algebra: STRAIGHT LINE: Linear Equation or Linear Function
A straight line is one of the most basic curves in both geometry and algebra. In algebra, the basic equation of a straight line is: y = mx + b where "m" is the slope of the straight line and "b" is the y-intercept of the straight line. Any equation that can be put into this (above) format is a straight line -and- all straight lines can (ultimately) be put into the format of this equation. This particular equation is known as the" "slope-intercept form" of a straight line. The slope "m" of a straight line is a measure of the tilt or angle of the straight line when graphed in a rectangular coordinate system. Slope is defined as "rise over run." If any two points on the straight line are picked ... say (x1,y1) and (x2,y2) ... and the rise is computed by taking the y-value of the second point minus the y-value of the first point (y2 - y1) ... and this is divided by the run which is computed by taking the x-value of the second point minus the x-value of the first point (x2 - x1) ... the slope of the line will be found. Slope = m = ( y2 - y1) / ( x2 - x1 ) = (change in y) / ( change in x ) The slope of a straight line is always constant ... no matter what two points are selected to compute it. A negative slope indicates that the line is moving downward to the right and decreasing; a zero slope indicates a line is perfectly horizontal and level; a positive slope indicates that a line is moving upward to the right and increasing; no slope (infinite slope is some textbooks) indicates that a line is perfectly vertical. The Y-Intercept of a straight line is the intersection point of the straight line with the y-axis when graphed in a rectangular (Cartesian) coordinate system. It is technically that point which corresponds to an x-value of 0 (zero) ... the matched y-value in the equation when "x" is set equal to zero. Example: Given the equation 2x + 3y = 9, describe the equation. Procedure ... solve the equation for the variable "y" ... subtract "2x" from both sides, divide both sides by "3" ... yielding: y = -2x / 3 + 3 -or- y = (-2/3)x + 3 This equation is clearly of the form "y = mx + b" and, therefore, must be a straight line. The slope is the x-coefficient which is -2/3; the y-intercept is the constant term 3. Description: The equation is a straight line. The slope of the straight line is -2/3. The y-intercept of the straight line is 3. The line is moving downward to the right (decreasing) when graphed in a coordinate system. The domain of the curve (the x values which make sense for the equation) is all real numbers. The domain of the curve (the y-values produced by substituting in all allowed x-values) is all real numbers. To graph the curve, you would make an x-y coordinate system and then locate the y-intercept first. From the y-intercept you would mark off the slope ... going down 2 and over to the right 3 ... locating a second point of the line. You would join the y-intercept and this second point with a straight line. Other Info: Two straight lines are parallel if and only if they have the same slopes. Two straight line are perpendicular if and only if they have opposite reciprocal slopes (i.e: if the slope of the first is -2/3 then the other is perpendicular if it has a slope of +3/2).
Algebra: Solving Inequalities
There are a number of different methods used for solving inequalities. These methods are outlined in different math textbooks. The method used in this presentation demonstrates one of these methods. Concept & Philosophy: To solve an inequality, find out where equality occurs. If one can determine where equality occurs ... then you also can determine where inequality occurs. The only issue becomes "which region/interval" of inequality is that being sought by/in the original problem. An example: Imagine you need to solve the inequality ... x2 + 3x - 10 < 0 Step 1: Change the inequality to an equality and solve this equality. (You should drop the inequality symbol and replace it with an equality symbol ... and then solve the resulting equation using whatever methods seem appropriate.) x2 + 3x - 10 = 0 ( x + 5 ) ( x - 2 ) = 0 x = -5 or x = +2 Step 2: Make an "x" number line locating the points of equality found above (in step 1). Put a "hard dot" (filled-in dot) on the points if the original problem allowed equality, Put an "open circle" around the points if the original problem was a "pure" inequality (allowing for no potential of equality). In this case ... locate the points and put open circles around them. Step 3: Examine the number line from step 2. It will be divided into intervals ... each of which represents a region of inequality. In this instance, there are three regions of inequality: all values to the left of and less than -5, the region between -5 and 2, and then the region comprised of all values to the right of and greater than 2. The issue you face: which of these three regions represent the one(s) required by the original problem ... the inequality stated in the original problem. To determine which (if any) of the regions work ... pick a value from each of the regions and test this value back in the original inequality. If the value makes the original inequality true, then ALL values in the interval work. If the test value makes the original inequality false, then NO value in the interval work. In this case: From left most interval try x = -10: (-10)2 + 3*(-10) - 10 = 60 which is NOT less than "0". NO VALUE IN LEFT MOST INTERVAL WORK!!! From the middle interval try x = 1: 12 + 3*1 - 10 = -6 which IS less than "0". ALL VALUES IN THE MIDDLE INTERVAL SATISFY THE ORIGINAL & WORK!!! From the right most interval try x = 4: 42 + 3*4 - 10 = 18 which is NOT less than "0". NO VALUE IN RIGHT MOST INTERVAL WORK!!! Step 4: Shade the region (interval) of the number line that makes the original inequality true ... then express this solution as an algebraic statement and as an interval. In this case, the algebraic solution is -5 < x < 2 and the interval solution is (-5,2). Other: If dealing with a rational (fractional) expression, one must use all values which cause a zero in the denominator. In general, any values that cause a function to be undefined are critical values ... and must be considered and graphed in step 2 above.
Algebra: Polynomials
(a+b) 2 = a 2 + 2ab + b 2 (a+b)(c+d) = ac + ad + bc + bd a 2 - b 2 = (a+b)(a-b) (Difference of squares) a 3 b 3 = (a b)(a 2 ab + b 2) (Sum and Difference of Cubes) x 2 + (a+b)x + ab = (x + a)(x + b) If ax 2 + bx + c = 0, then x = ( -b (b 2 - 4ac) ) / 2a (The Quadratic Formula for solving second degree equations ... see below.) Solving Quadratic Equations There are two common methods used to solve 2nd degree (quadratic) equations: factoring or the quadratic formula. In both cases, the equation being solved must be put into standard form (ax2 + bx + c = 0) as a first preliminary step. Standard form requires that all variables and numbers be simplified and grouped onto one side of the equation -and- set equal to zero on the other side of the equation. For example, if the original equation is: x(x + 2) = 8 ... the standard form would be: x2 + 2x - 8 = 0 At this point ... the most common solution method is to attempt to factor the equation. The equation x2 + 2x - 8 = 0 (from above) can be factored: ( x - 2 ) ( x + 4 ) = 0 Assuming the equation can be factored, one then cross equates each of the factors to zero ... and solves the sub-equations: x - 2 = 0 -or- x + 4 = 0 x = 2 -or- x = -4 If the equation can NOT be solved by factoring, then the quadratic formula is used. The "a," "b," and "c" values for the equation are identified and then "plugged into" the quadratic formula.In our example (above): a=1, b=2, and c=-8. Note One: In the quadratic formula it is assumed that the ENTIRE top of the equation is divided by the entire bottom. Note Two: It is assumed that the "b2 - 4ac" must be computed prior to taking the square root. Note Three: If "b2 - 4ac" is positive, then the equation has two real number roots (solutions). If "b2 - 4ac" is zero, then the equation has only one real number root (solution). If "b2 - 4ac" is negative, then the equation has NO real number roots (solutions) ... it has two complex number solutions.
Algebra: Examples of Algebra Graphs
Examples of Algebra Graphs (Hoxie H.S., Kansas)
Algebra: Equation & Expression Solver
Algebraic Equation & Expression Solver (Hoxie H.S., Kansas)
Geometry: Formulas
Geometric Formulas (Hoxie H.S., Kansas)
Trigonometry: Identities
Trigonometric Identities (Hoxie H.S., Kansas)
Calculus: Vectors
Calculus: Vectors (Hoxie H.S., Kansas)
Calculus: Symbols
Calculus: Symbols (Hoxie H.S., Kansas)