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Published:
January 14, 2019

Algebra II For Dummies

Overview

Algebra II For Dummies, 2nd Edition (9781119543145) was previously published as Algebra II For Dummies, 2nd Edition (9781119090625). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.

Your complete guide to acing Algebra II 

Do quadratic equations make you queasy? Does the mere thought of logarithms make you feel lethargic? You're not alone! Algebra can induce anxiety in the best of us, especially for the masses that have never counted math as their forte. But here's the good news: you no longer have to suffer through statistics, sequences, and series alone. Algebra II For Dummies takes the fear out of this math course and gives you easy-to-follow, friendly guidance on everything you'll encounter in the classroom and arms you with the skills and confidence you need to score high at exam time.

Gone are the days that Algebra II is a subject that only the serious 'math' students need to worry about. Now, as the concepts and material covered in a typical Algebra II course are consistently popping up on standardized tests like the SAT and ACT, the demand for advanced guidance on this subject has never been more urgent. Thankfully, this new edition of Algebra II For Dummies answers the call with a friendly and accessible approach to this often-intimidating subject, offering you a closer look at exponentials, graphing inequalities, and other topics in a way you can understand. 

  • Examine exponentials like a pro
  • Find out how to graph inequalities
  • Go beyond your Algebra I knowledge
  • Ace your Algebra II exams with ease

Whether you're looking to increase your score on a standardized test or simply succeed in your Algebra II course, this friendly guide makes it possible.

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About The Author

Mary Jane Sterling was a lecturer in mathematics for more than 35 years, teaching courses in algebra, calculus, and linear programming. She is the author ofAlgebra I For Dummies, Trigonometry For Dummies, Algebra Workbook For Dummies, and Trigonometry Workbook For Dummies.

Sample Chapters

algebra ii for dummies

CHEAT SHEET

Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!Algebraic equations for multiplying binomialsIn algebra, multiplying binomials is easier if you recognize their patterns.

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Articles from
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Numbers are your friends. You may not always agree with this, especially if your weight is too great or your bank account is too small. But numbers have been around for centuries and have helped humans keep track of what is going on in their lives. What are the most favorite or most famous numbers? Try these out for size!
In algebra, multiplying binomials is easier if you recognize their patterns. You multiply the sum and difference of binomials and multiply by squaring and cubing to find some of the special products in algebra. See if you can spot the patterns in these equations: Sum and difference: (a + b)(a – b) = a2 – b2
Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!Algebraic equations for multiplying binomialsIn algebra, multiplying binomials is easier if you recognize their patterns.
In algebra, you use permutations to count the number of subsets of a larger set. Use permutations when order is necessary. With combinations, you can count the number of subsets when order doesn't matter. The formulas you need are
Logarithms help you add instead of multiply. The algebra formulas here make it easy to find equivalence, the logarithm of a product, quotient, power, reciprocal, base, and the log of 1.
You can find solutions for quadratic equations by factoring, completing the square, guessing, or everyone's favorite — using the quadratic formula. The best thing about the quadratic formula (unlike the other methods) is that it always works.
Here are a few alphabet functions, called that because they are named using letters from the Greek alphabet. In algebra, a function is a rule or relation that is defined using various mathematical operators. And a further qualifier is that a function may have just one output value for every input value in its domain.
The numbers zero and one have special roles in algebra — as additive and multiplicative identities, respectively. You use identities in algebra when solving equations and simplifying expressions. You need to keep an expression equal to the same value, but you want to change its format, so you use an identity in one way or another: The additive identity is zero.
When you complete the square on an equation with both x's and y's, the result is a standard form of the equation for a conic section. You can then use this standard form to uncover more information about the conic section. Conic sections (circles, ellipses, hyperbolas, and parabolas) have standard equations that give you plenty of information about individual curves — where their centers are, which direction they go in, and so on.
The composition of functions is an algebraic operation in which you use one function as the input into another and perform the operations on that input function. You can perform the basic mathematical operations of addition, subtraction, multiplication, and division on the equations used to describe functions.
When the equation of a parabola appears in standard form, you have all the information you need to graph it or to determine some of its characteristics, such as direction or size. Not all equations come packaged that way, though. You may have to do some work on the equation first to be able to identify anything about the parabola.
When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle.The equation x2 + y2 + 6x – 4y – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables.
Named for Gabriel Cramer, Cramer’s Rule provides a solution for a system of two linear algebraic equations in terms of determinants — the numbers associated with a specific, square matrix.
Three famous problems confounded mathematicians for centuries: squaring the circle, trisecting the angle, and doubling the cube. These problems are quite accessible nowadays with modern computing and technology. But the ancients only had a compass and straightedge to work with, so these problems were pretty much unsolvable.
Algebra is all about graphing relationships, and the curve is one of the most basic shapes used. Here's a look at eight of the most frequently used graphs.
A useful method for solving algebraic equations that contain negative exponents is to factor out a negative greatest common factor, or GCF. For example, consider the equation 3x –3 – 5x –2 = 0. This equation has a solution that you can find without switching to fractions right away. In general, equations that have no constant terms — all the terms have variables with exponents on them — work best with this technique.
Like quadratic binomials, a quadratic trinomial can have as many as two solutions — or it may have one solution or no solution at all. If you can factor the trinomial and use the multiplication property of zero to solve for the roots, you're home free. The multiplication property of zero states that if the product of then at least one of the factors has to represent the number 0.
Factoring by grouping terms is a great method to use to rewrite a quadratic equation so that you can use the multiplication property of zero and find all the solutions. The main idea behind factoring by grouping is to arrange the terms into smaller groupings that have a common factor. You go to little groupings because you can't find a greatest common factor for all the terms; however, by taking two terms at a time, you can find something to divide them by.
In algebra, a one-sided limit tells you what a function is doing at an x-value as the function approaches from one side or the other. One-sided limits are restrictive, and work only from the left or from the right. When a rational function doesn't have a limit at a particular value, the function values and graph have to go somewhere.
A piecewise function consists of two or more function rules (function equations) pieced together (listed separately for different x values) to form one bigger function. A change in the function equation occurs for different values in the domain. For example, you may have one rule for all the negative numbers, another rule for numbers bigger than three, and a third rule for all the numbers between those two rules.
You can find the solution of a system of inequalities involving a line and a curve (such as a parabola), two curves, or any other such combination. You do this by graphing the individual equations, determining which side to shade for each curve, and identifying where the equations share the shading. For example, to solve the system make your way through the following steps: Graph the line y = x – 3 and determine which side of the line to shade by checking a test point (a random point that's clearly on one side or the other) to see if it satisfies the inequality.
Functions can have many classifications or names, depending on the situation and what you want to do with them. One very important classification is deciding whether a function is one-to-one. You can do this using graphing techniques called vertical and horizontal line tests. A function is one-to-one if it has exactly one output value for every input value and exactly one input value for every output value.
Life isn't always as easy as one-variable equations. Being able to solve an algebra equation for some variable when it contains more than one unknown can be helpful in many situations. If you're repeating a task over and over — such as trying different widths of gardens or diameters of pools to find the best size — you can solve for one of the variables in the equation in terms of the others.
There are so many different stories about the original discovery of the chessboard. And many of the stories also involve grains of oats, rice, or barley-corn and the ouster of a king. Which story was first? Which story is true? Was it a king in ancient Persia, a sultan in Turkey, a prince in Babylonia? Some hints even indicate that the basis of the story comes from the Bible.
Some algebraic equations that contain radicals call for more than one application of squaring both sides. For example, you usually need to square both sides more than once when you have three terms in the equation — two of them with radicals. For example, say you have to work with the equation Here's how you solve the problem: Move the radicals so that only one appears on each side.
Radicals can make equations very difficult to work with. Fortunately, it's possible to get rid of a radical by squaring both sides of the equation; for this method to work, the equation must be in the form The only problem arises when you end up with an extraneous root. Consider the non-equation –3 = 3. You know that the equation isn't correct, but what happens when you square both sides of this statement?
Working with radicals can be troublesome, but these equivalences keep algebraic radicals from running amok. Finding the root of product or quotient or a fractional exponent is simple with these formulas; just be sure that the numbers replacing the factors a and b are positive.
The problem with fractions in linear algebraic equations is that they aren't particularly easy to deal with. For example, they always require common denominators before you can add or subtract them. The best way to deal with linear equations that involve variables tangled up with fractions is to get rid of the fractions.
When you need to simplify an algebraic equation that involves logarithms, one way to make the equation much nicer is by using the basic exponential/logarithmic equivalence If you're asked to evaluate log9 3, for example (or if you have to change it into another form), you can write it as an equation, log9 3 = x, and use the equivalence: 9x = 3.
The difference quotient shows up in most high school Algebra II classes as an exercise you do after your instructor shows you the composition of functions. You perform this exercise because the difference quotient is the basis of the definition of the derivative. The difference quotient allows you to find the derivative, which allows you to be successful in calculus (because everyone wants to be successful in calculus, of course).
If you ever have to solve a quadratic-like trinomial, you're in luck; this type of equation is a perfect candidate for factoring and then for the application of the multiplication property of zero. A quadratic-like trinomial is a trinomial of the form ax2n + bxn + c = 0. The power on one variable term is twice that of the other variable term, and a constant term completes the picture.
Proportions are nice to work with when you're solving rational equations, because you can eliminate the fractions or change them so that they feature better denominators. A proportion is an equation in which one fraction is set equal to another. For example, the equation is a proportion. When you have the proportion the following are also true: the cross-products, ad and bc, are equal, giving you ad = bc.
Any equation with one or more rational terms (or fractions) is a rational equation. Although these equations aren't always easy to handle, you can turn them into quadratic equations by finding their least common denominator, or LCD. To solve a rational equation with the LCD, you find a common denominator, write each fraction with that common denominator, and then multiply each side of the equation by that same denominator to get a nice quadratic equation.
When solving rational inequalities, you use the same steps as for any quadratic inequality. By adding the sign-line method, you can also learn whether the different factors in each interval are positive or negative. Begin with these steps: Move all the terms to one side of the inequality sign. Factor, if possible.
In algebra, solving a quadratic inequality is very similar to solving a quadratic equation. The difference is that with quadratic equations, you set the expressions equal to zero, but with inequalities, you're interested in what's on either side of the zero (positives and negatives). To solve a quadratic inequality, you follow these steps: Move all the terms to one side of the inequality sign.
Conics are curved algebraic forms that come from slicing a cone with a plane. Use these equations to graph algebraic conics, such as circles, ellipses, parabolas, and hyperbolas:
The multiplication property of zero is really useful for doing algebra. Of course, you may be thinking that multiplying by zero is no big deal. After all, zero times anything is zero, right? Well, that's exactly why it's a big deal. Read on... You can use the multiplication property of zero when solving equations.
You can use the base rule to solve algebraic equations with different bases, as long as the bases are related to one another by being powers of the same number. If you have an equation written in the form bx = by, where the same number represents the bases b, the following rule holds: You read the rule as follows: "If b raised to the xth power is equal to b raised to the yth power, that implies that x = y.
Algebra can help you add a series of numbers (the sum of sequences) more quickly than you would be able to with straight addition. Adding integers, squares, cubes, and terms in an arithmetic or geometric sequence is simple with these algebraic formulas:
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