What Is an Identity?
An identity is a special type of equation. It includes at least one variable and an identity is true for all values of the variable.
Let’s look at an example. Here is an equation.
No matter which value of x you use, the equation is true. Here are some examples.
In each case the equation is true. Try other values for x. In all cases, the result is a true equation.
Conditional Equations
Now let’s look at another equation.
This equation is true only for x = 8. Any other values of x results in a false equation. This is an example of a conditional equation.
The difference between identities and conditional equations is the number of solutions to the equation.
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The Simplest Identity
The simplest identity is when a variable equals itself.
This is an example of the Reflexive Property of Equality. In this equation, no matter what values of x are used, the equation is true.
Most identities are equations that have different expressions on each side of the equals sign. Let’s look at some examples.
Binomial Identities
Identities are a way of rewriting an expression, especially when solving an equation. Some of the most common identities you’ll come across are binomial identities, which is an expression with two terms.
Here is the square of a binomial of the form (x + a):
Here is the square of a binomial of the form (x - a):
You should also learn how to recognize when a polynomial of degree 2 can be written as a binomial squared. Let’s look at an example.
This expression
Can also be written this way:
Using the identity, you can the rewrite the expression to get this equation.
Learning these identities is helpful, especially when solving quadratic equations. Let’s look at an example.
See how using these identities are helpful in solving quadratic equations? Learning these identities will help you recognize how to factor certain types of quadratic equations.
Difference of Squares
Another helpful identity is known as the difference of squares. The identity looks like this:
The term on the left is the difference of two terms that are squared. The term on the right is the product of two binomials.
Use this identity to factor certain quadratic expressions. Here’s an example.
The expression
Can be rewritten this way:
See how this is a difference of squares? It can be rewritten as the product of two binomials.
Use this identity to help factor certain types of quadratic expressions.
Sum and Difference of Cubes
Another helpful pair of identities are the sum of cubes and the difference of cubes. The identities looks like this:
Here is an example.
This expression
Can be written this way:
Using the difference of cubes identity we get this equation:
Trigonometric Identities
There is a long list of trigonometric identities. Here are some examples.
