Algebra Articles

Cheat Sheet / Updated 04-28-2022

Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).

Cheat Sheet / Updated 04-19-2022

The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.

Cheat Sheet / Updated 04-12-2022

Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.

Cheat Sheet / Updated 03-28-2022

Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice. The only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.

Cheat Sheet / Updated 03-14-2022

To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.

Cheat Sheet / Updated 02-24-2022

Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.

Cheat Sheet / Updated 01-21-2022

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!

Article / Updated 12-21-2021

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. Just follow these steps: Change the order of the terms so that the x's and y's are grouped together and the constant appears on the other side of the equal sign. Leave a space after the groupings for the numbers that you need to add: x2 + 6x _____ + y2 – 4y _____ = 3 Complete the square for each variable, adding the number that creates perfect square trinomials. In the case of the x's, you add 9, and with the y's, you add 4. Don't forget to also add 9 and 4 to the right: x2 + 6x + 9 + y2 – 4y + 4 = 3 + 9 + 4 When it's simplified, you have x2 + 6x + 9 + y2 – 4y + 4 = 16 Factor each perfect square trinomial. The standard form for the equation of this circle is (x + 3)2 + (y – 2)2 = 16. The circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure below shows you the way. With the center, radius, and a compass, you too can sketch this circle.

Article / Updated 07-14-2021

To add or subtract with powers, both the variables and the exponents of the variables must be the same. You perform the required operations on the coefficients, leaving the variable and exponent as they are. When adding or subtracting with powers, the terms that combine always have exactly the same variables with exactly the same powers. These rules are true for multiplying and dividing exponents as well. Example 1: x + x + x = 3x Because the variables are the same (x) and the powers are the same (there are no exponents, so the exponents must be 1), you can add the variables. Example 2: Because the variables are the same (x) and the powers are the same (2), you can perform the required operations on the variables. Example 3: Although the variables are the same (x), the powers are not the same (1, 2, 3, and 4). You can't simplify these terms because only the variables are the same, and both the variables and the powers need to be the same. Example 4: Sometimes not all of the variables and powers will be the same — you may encounter a problem that has several groups of variables and powers that are not the same. In this case, you only add or subtract terms whose variables and powers are the same. (Notice that the exponents are listed in order from highest to lowest. This is a common practice to make answers easy to compare.)

Article / Updated 07-14-2021

Combining square roots Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” Finding square roots is a relatively common operation in algebra, but working with and combining the roots isn’t always so clear. Expressions with radicals can be multiplied or divided as long as the root power or value under the radical is the same. Expressions with radicals cannot be added or subtracted unless both the root power and the value under the radical are the same. When you find square roots, the symbol for that operation is called a radical. The root power refers to the number outside and to the upper left of the radical. If there is no number, you assume that the root power is 2. When it comes to combining radicals, there are just a couple of simple rules to remember: Addition and subtraction can be performed if the root power and the value under the radical are the same. Examples: These radicals cannot be combined because the operation is addition, and the value under the radical is not the same: These radicals can be combined because the root power and the numbers under the radical are the same: These radicals cannot be combined because the operation is subtraction, and the root power isn’t the same: Multiplication and division can be performed if the root powers are the same. Examples: These radicals can be combined because the operation is multiplication, and the root power is the same: These radicals can be combined because the operation is division, and the root power is the same: These radicals cannot be combined because the operation is division, and the root power isn’t the same: Converting square roots to exponents Finding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical. When you find square roots, the symbol for that operation is a radical, which looks like this: When changing from radical form to fractional exponents, remember these basic forms: The nth root of a can be written as a fractional exponent with a raised to the reciprocal of that power. When the nth root of is taken, it’s raised to the 1/n power. When a power is raised to another power, you multiply the powers together, and so the m (otherwise written as m/1) and the 1/n are multiplied together. Use fractions in the powers to indicate that the expression stands for a root or a radical. Here are some examples of changing radical forms to fractional exponents: When raising a power to a power, you multiply the exponents, but the bases have to be the same. Because raising a power to a power means that you multiply exponents (as long as the bases are the same), you can simplify the following expressions: Leave the exponent as 9/4. Don’t write it as a mixed number. The following example can’t be combined because the bases are not the same: