Purplemath

Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " …

Purplemath. So far, we've dealt with each type of asymptote separately, giving one page to each type, kind of like your textbook probably does, giving one section to each type. But on the test, the questions won't specify which type of asymptote you'll need to find. You can solve this "space" problem by using negative numbers. The "whole" numbers start at zero and count off to the right; these are the positive integers. The negative integers start at zero and count off to the left: Note the arrowhead on the far right end of the number line above. That arrow tells you the direction in which the … Use completing the square to solve x2 − 4x − 8 = 0. As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the equation in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that ready-for-square-rooting form, so I can solve. Evaluate 6!. A factorial is just a product. To "evaluate" a factorial is simply to multiply it out. In this case, they're wanting me to "take the factorial of" 6. This means that I need to multiply all the whole numbers from 1 through 6, inclusive. My work is pretty simple: 1×2×3×4×5×6 = 720. This value is all they're looking for, so my ... Also, this hyperbola's foci and vertices are to the left and right of the center, on a horizontal line paralleling the x -axis. From the equation, clearly the center is at (h, k) = (−3, 2). Since the vertices are a = 4 units to either side, then they are at the points (−7, 2) and at (1, 2). The equation a2 + b2 = c2 gives me:You should know the formula for the circumference C and area A of a circle, given the radius r: Acir = π r2. Ccir = 2π r. (" π " is the number approximated by 3.14159 or the fraction 22/7) Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the radius is just halfway across.Then the GCF is 2 × 3 × 5 × 7 = 210. On the other hand, the Least Common Multiple, the LCM, is the smallest (that is, the "least") number that both 2940 and 3150 will divide into. That is, it is the smallest number that contains both 2940 and 3150 as factors, the smallest number that is a *multiple* that is common to both these values. Therefore, it will be the …

Pre-algebra and algebra lessons, from negative numbers through pre-calculus. Grouped by level of study. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. Purplemath. Unlike the examples on the previous page, nearly all polynomial divisions do not "come out even"; usually, you'll end up with a remainder. Divide 3x 3 − 5x 2 + 10x − 3 by 3x + 1; I start with the long-division set-up: Looking only at the leading terms, I divide 3x 3 by 3x to get x 2. This is what I put on top:Purplemath What is a circle? A circle is a geometrical shape. It is defined as having a center, and being the set of all points that are a certain fixed distance from that center. (The fixed distance is called the radius of the circle.) The circle is not of much use in algebra since the equation of a circle isn't a function.The basic metric units are meters (for length), grams (for mass or weight), and liters (for volume). And the different units convert into one another rather nicely, with one milliliter equalling one cubic centimeter (where one Cubic Centimeter is the "cc" of medical shows on television) and one gram being the mass (or weight) of one cc …Use completing the square to solve x2 − 4x − 8 = 0. As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the equation in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that ready-for-square-rooting form, so I can solve.In the above example, the degree on the denominator (namely, 2) was bigger than the degree on the numerator (namely, 1), and the horizontal asymptote was y = 0 (that is, it was the x-axis).This property is always true: If the degree on x in the denominator is larger than the degree on x in the numerator, then the …Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms.

Now I can solve each factor by setting each one equal to zero and solving the resulting linear equations: x + 2 = 0 or x + 3 = 0. x = −2 or x = − 3. These two values are the solution to the original quadratic equation. So my answer is: x = −3, −2. Lessons and Tutoring - Reviews. The reviews below refer to free (or free-to-try) off-site tutoring and instructional resources. To access the Purplemath lessons and tutoring forums, please use the links to the right. For paid in-home tutoring, please try here. algebra.help: This site has lessons on basic algebra topics and techniques, …Purplemath is a website that provides free math lessons and resources for students and teachers. It started in 1998 as a personal web site by Elizabeth Stapel, and has grown to …For the three-sevenths fraction, the denominator needed a factor of 5, so I multiplied by \frac {5} {5} 55, which is just 1. We can use this same technique to rationalize radical denominators. I could take a 3 out of the denominator of my radical fraction if I had two factors of 3 inside the radical.

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The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (that is, of multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x − 2) is a binomial, 10 is a rather large exponent, and (3x − 2)10 would be very painful to multiply out by hand.Logarithms are inverse functions (backwards), and logs represent exponents (concept), and taking logs is the undoing of exponentials (backwards and a concept). And this is a lot to take in all at once. Yes, in a sense, logarithms are themselves exponents. Logarithms have bases, just as do exponentials; for instance, log5(25) …Lessons and Tutoring - Reviews. The reviews below refer to free (or free-to-try) off-site tutoring and instructional resources. To access the Purplemath lessons and tutoring forums, please use the links to the right. For paid in-home tutoring, please try here. algebra.help: This site has lessons on basic algebra topics and techniques, study ...Purplemath. Once you've learned the basic keywords for translating word problems from English into mathematical expressions and equations, you'll be presented with various English expressions, and be told to perform the translation. Don't view the lists of keywords as holy writ, handed down from on high. Instead, use these lists … Using these numbers, I can split the middle −13x term into the two terms −9x and −4x, and then I can factor in pairs: 6 x2 − 13x + 6. = 6 x2 − 9x − 4x + 6. = 3 x (2 x − 3) − 2 (2 x − 3) = (2x − 3) (3x − 2) The factoring method in the last two examples above — in particular, the part where I picked two numbers for ...

Purplemath. When you're graphing (or looking at a graph of) polynomials, it can help to already have an idea of what basic polynomial shapes look like. One of the aspects of this is "end behavior", and it's pretty easy. We'll look at some graphs, to find similarities and differences. First, let's look at some polynomials of even degree ... Purplemath. You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that x 3 × x 5 equals x 8 because you can add the exponents. There are similar rules for logarithms. (I'll provide proofs for each of the rules. You almost certainly don't need to know …Also, this hyperbola's foci and vertices are to the left and right of the center, on a horizontal line paralleling the x -axis. From the equation, clearly the center is at (h, k) = (−3, 2). Since the vertices are a = 4 units to either side, then they are at the points (−7, 2) and at (1, 2). The equation a2 + b2 = c2 gives me:Purplemath. A very common class of "proportions" exercise is that of finding the height of something very tall by using the daytime shadow length of that same thing, its shadow being measured horizontally along the ground. In such an exercise, we use the known height of something shorter, along with the length of that shorter … Free math problem solver answers your algebra homework questions with step-by-step explanations. Purplemath. You've already learned the basic trig graphs. But just as you could make the basic quadratic, y = x2, more complicated, such as y = − (x + 5)2 − 3, so also trig graphs can be made more complicated. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. Purplemath. A ratio is one thing or value compared with or related to another thing or value; it is just a statement or an expression, and can only perhaps be simplified or reduced. On the other hand, a proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved. Purplemath. To be honest, solving "by graphing" is a somewhat bogus topic. The basic idea behind solving by graphing is that, since the (real-number) solutions to any equation (quadratic equations included) are the x-intercepts of that equation, we can look at the x-intercepts of the graph to find the solutions to the corresponding …Purplemath. In addition to finding lines (axes) of symmetry, you can also look for points of symmetry. A point of symmetry is a point that represents a "center" of sorts for the figure. For any line that you draw through the point of symmetry, if this line crosses the figure on one side of the point, the line will also cross the figure on the ...Purplemath. When you work with angles in all four quadrants, the trig ratios for those angles are computed in terms of the values of x, y, and r, where r is the radius of the circle that corresponds to the hypotenuse of the right triangle for your angle. In the drawing below, the angle ends in the second quadrant, as indicated by the … Purplemath. So far, we've dealt with each type of asymptote separately, giving one page to each type, kind of like your textbook probably does, giving one section to each type. But on the test, the questions won't specify which type of asymptote you'll need to find.

Purplemath Linear programming is the process of taking various linear inequalities (called "constraints") relating to some situation, and finding the best value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the optimal production levels for maximal profits …

Purplemath. An important category of percentage exercises is markup and markdown problems. For these, you calculate the markup or markdown of the price or cost in absolute terms (you find by how much the price or cost changed), and then you calculate the percent change relative to the original value. So they're really …Purplemath. Straight-line equations, or "linear" equations, graph as straight lines, and have simple variable expressions with no exponents on them. If you see an equation with only x and y − as opposed to, say x 2 or sqrt(y) − then you're dealing with a straight-line equation.. There are different types of "standard" formats for …The foci are side by side, so this hyperbola's branches are side by side, and the center, foci, and vertices lie on a line paralleling the x -axis. So the y part of the equation will be subtracted and the a2 will go with the x part of the equation. The center is midway between the two foci, so the center must be at (h, k) = (−1, 0). Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7. The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5. The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an ... Purplemath Linear programming is the process of taking various linear inequalities (called "constraints") relating to some situation, and finding the best value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the optimal production levels for maximal profits … Purplemath. So far, we've dealt with each type of asymptote separately, giving one page to each type, kind of like your textbook probably does, giving one section to each type. But on the test, the questions won't specify which type of asymptote you'll need to find. Compound (or compounded) interest is interest that is earned on interest. If you invest $300 in a compound-interest fund for two years at 10% interest annually, you will earn $30 for the first year, but then you will earn 10% of $330 (or $33) for the second year, for a total of $63 in interest. Content Continues Below.

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24 trailing zeroes in 101! This reasoning, of finding the number of multiples of 51 = 5, plus the number of multiples of 52 = 25, etc, extends to working with even larger factorials. Find the number of trailing zeroes in the expansion of 1000! Okay, there are 1000 ÷ 5 = 200 multiples of 5 between 1 and 1000. The next power of 5, …An identity is a tautology, an equation or statement that is always true, no matter what you plug in for the variable. Learn how to prove an identity using logical steps and notation, …You should know the formula for the circumference C and area A of a circle, given the radius r: Acir = π r2. Ccir = 2π r. (" π " is the number approximated by 3.14159 or the fraction 22/7) Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the radius is just halfway across. The Purplemath lessons have been written so that they may be studied in whatever manner the student finds most useful. Different textbooks cover different topics in different orders. The Purplemath lessons try not to assume any fixed ordering of topics, so that any student, regardless of the textbook being, may benefit. Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " …Purplemath What are a number's "factors"? "Factors" are the whole numbers you multiply to get another whole number. For instance, factors of 15 are 3 and 5, because 3 × 5 = 15. Some numbers have more than one factorization (more than one way of being factored). For instance, 12 can be factored as 1 ×12, 2 × 6, and also …Purplemath. You may be asked about the "correlation", if any, displayed within a particular scatterplot. The word orrelation can be used in at least two different ways: to refer to how well an equation matches the scatterplot, or to refer to the way in which the dots line up. If you're asked about "positive" or "negative" correlation, …Purplemath offers free algebra lessons, homework guidelines, and study skills survey for students of all levels and ages. Learn how to prepare for tests, avoid common mistakes, …Purplemath. The next level of this type of log equation may require a calculator to solve. You'll still find the solution using algebra, but they'll be wanting a decimal approximation for non-"nice" values, which will require "technology". An example would be: Solve ln(x) = 3, giving your answer accurate to three decimal places. Here are some suggestions to help you prepare for the ALEKS math test. Start with an ALEKS math practice test. Create a plan to master the topics you need to learn. Follow a daily routine of ALEKS math test prep. Evaluate your learning. Get ALEKS math help with any difficult concepts. Trust your ability to achieve a good score. Purplemath. On the previous page, we saw how we could expand the context of the trigonometric ratios from the geometric one of right triangles to the algebraic one of angles being based at the origin and using angles of any measure.. This disconnects the trig ratios from physical constraints, allowing the ratios to become useful in … ….

Trigonometric Identities. Unit Circle. Find a clear explanation of your topic in this index of lessons, or enter your keywords in the Search box. Free algebra help is here! Pre-algebra and algebra lessons, from negative numbers through pre-calculus. Grouped by level of study. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. Purplemath. A very common class of "proportions" exercise is that of finding the height of something very tall by using the daytime shadow length of that same thing, its shadow being measured horizontally along the ground. In such an exercise, we use the known height of something shorter, along with the length of that shorter …Purplemath. Once you've learned the basic keywords for translating word problems from English into mathematical expressions and equations, you'll be presented with various English expressions, and be told to perform the translation. Don't view the lists of keywords as holy writ, handed down from on high. Instead, use these lists …Note this common technique: In the "n = k + 1" step, it is usually a good first step to write out the whole formula in terms of k + 1, and then break off the "n = k", so you can replace it with whatever assumption you made about n = k in the assumption step.Then you manipulate and simplify, and try to rearrange things to get the RHS …Purplemath Linear programming is the process of taking various linear inequalities (called "constraints") relating to some situation, and finding the best value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the optimal production levels for maximal profits … Purplemath What is an identity? In mathematics, an "identity" is an equation which is always true, regardless of the specific value of a given variable. An identity can be "trivially" true, such as the equation x = x or an identity can be usefully true, such as the Pythagorean Theorem's a 2 + b 2 = c 2 Purplemath. Variation problems aren't hard once you get the hang of the lingo. The only real difficulty is learning the somewhat specialized vocabulary and the techniques for this classification of problems. Variation problems involve fairly simple relationships or formulas, involving one variable being equal to one term. Purplemath, [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1], [text-1-1]