EQUATIONS

Solving equations is a critical part of Algebra. An equation is a mathematical statement that includes an equal sign to separate two algebraic expressions. There are many types included in a typical Algebra course. Most types are included in the video content below. There are many ways to solve equations, and each type requires different techniques. To be successful in an Algebra course, students must be able to solve equations.

Linear

Learn to solve various levels of linear equations. Types include solving multi-step, variables on both sides, utilizing the distributive property, and learning how to clear an equation of fractions. Linear equations have at most one solution. It is possible to have infinite solutions, this occurs if the variable cancels out and you are left with a true statement.  It is also possible to have no solutions, this occurs if the variable cancels out and you are left with a false statement. Examples of both infinite solutions and no solutions are included.
 

Quadratic

Quadratic equations contain a  variable that is squared.  The most common way to solve a quadratic equation is to use the quadratic formula. Other ways of solving quadratic equations include factoring, completing the square, and the square root method.  Factoring can only be used if the solutions are rational.  The square root method can only be used if there is no linear (x) term, and the variable is only squared. Completing the square and the quadratic formula can always be used, regardless of the type of solution. Quadratics can have at most 2 solutions.  It is possible to have no real solutions, a.k.a. imaginary solutions, in quadratic equations. Depending on the level in Algebra, students will either list the imaginary solutions, or put no real solutions as the answer. 

Solve by Factoring Ex 1
Solve by Factoring Ex 2
Solve by Factoring Ex 3
Solve by Square Root Method
Solve Using the
Quadratic Formula Ex 1
Quadratic Formula
No Real Solutions

Radical 

Radical equations contain an algebraic expression inside of a radical.  Radical equations include square root, cube roots, and up to nth roots.  There are two types of radical equations that are typically solved. Radical = radical equations have radicals on both sides.  In order to solve this type, both sides of the equation are taken to the power that would clear both radical terms. The other type of radical equations have a term that is not under the radical.  To solve this type, you would need to isolate the radical term, and take both sides to the root that gets rid of the radical.  It is possible to have extraneous solutions (extra solutions that would work if it is quadratic, but don't because we are only looking for the positive root), so it is important to always check solutions. An alternative way of writing a radical equation is to use rational exponents.  Rational exponents are exponents with fractions.

Basic Radical
Cube Root
Radicals on Both Sides
Radical - No Real Solutions
Radical- TI-84 used to Check Solution
Results in a Quadratic Equation
Rational Exponent
Rational Exponents in Quadratic Form Using a U-substitution

Polynomial Equations

Polynomial equations have at least one x term, and can go up to nth degree. This section will contain polynomial equations that are degree three and higher, since there is a separate section for quadratics and linear equations (which are special types of polynomial equations.) There are various methods of solving polynomial equations.  The highest degree of the polynomial will give the maximum number of solutions to the equation. There are only a couple of examples currently, but more will be added soon.

6th degree in Quadratic Form Using U-Substitution
In Quadratic Form Using U-Substitution

Rational Equations

Rational equations contain the variable term in the denominator.  The easiest way to solve these is to multiply both sides by the lowest common denominator in order to clear the fractions. Rational equations can range from very basic to very difficult. These can result in linear, quadratic or polynomial equations that have to be solved after clearing fractions. These also can contain extraneous solutions.  An extraneous solution in rational equations is a solution that when plugged back into the original expression result in a zero in the denominator.  It is very important to check solutions.  Various levels are covered.

Basic Rational Equation
Example with No Solution
Solve Using Cross Multiplication
Solve Using Factoring to find LCD-Linear
Solve Using Factoring to find LCD-Quadratic
Solve Using U-Substitution

Absolute Value Equations

Absolute value equations contain an algebraic expression inside of an absolute value.  When solving this type, it is important to remember the definition of absolute value.  Absolute value is the distance from zero.  Distance is always positive, and except for zero, there are always two numbers that are equal distances from zero.  For these, you want to first get the absolute value expression alone on one side, then set the expression equal to the positive and negative of the opposite side, then solve the resulting equations.

Absolute Value Ex 1
Absolute Value Ex 2
Absolute Value - No Solution

Logarithmic Equations

Logarithmic equations are equations containing logarithms. Traditionally, there are two types,
those that have logarithms on both sides and those that do not. Solving logarithmic equations is very helpful in an Algebra course.

Solve Logarithmic Equation Ex 1