Techniques for Solving Equations

Look at this equation.

This is an example of a conditional equation. This means that there are only a limited number of values of x that make this equation true. For example, watch what happens when x = 0 and x = 1.

Notice how each value of x results in a false equation? For what values of x is this equation true?

Answering this question involves solving an equation.

How Do You Solve an Equation?

As you’ve seen, the first step in solving an equation is to start with a conditional equation. The last step is to show the solution(s) to the equation. The solution process looks something like this.

 

What about the steps in the middle?

One thing to be aware of is that throughout the solution process, the equality in the equation is maintained. 

Throughout the process, the original equation is rewritten, but each version is equivalent to the original equation.

How is this possible?

Properties of Equality

To go from the original equation to the solution, use the Properties of Equality to rewrite the equation to get to the solution. This table summarizes these properties.

For many equations, a combination of these properties will get you to the solution of the equation. Let’s look at an example.

This solution process show five separate, but equivalent equations. Notice the use of the Subtraction Property of Equality and the Division Property of Equality. 

Here’s an example of a solution using the Addition Property of Equality and the Multiplication Property of Equality.

Equations with Two Solutions

Some polynomial functions, usually of degree 2, have two solutions. To solve a quadratic equation algebraically, you still use the Properties of Equality. For some quadratics, you also need to use the Square Root Property.

 

Case 1:

Let’s look at an example.

Notice that this example has a = 1 and results in two solutions. Now let’s look at a case where a > 1.

 

Case 2:

Some quadratic equations that you come across will have a quadratic term and a linear term. For these you would use the properties shown earlier, plus a few more. With this form you can factor an x-term. This relies on using the Distributive Property and the Reverse of the Distributive PropertyOnce factored, use the Zero Product Property.

Let’s look at an example.

 

For this quadratic the solutions are x = 0 and x = 2. With these types of quadratics, one of the solutions is always zero. Here’s an example where a > 1.

For this quadratic the solutions are x = 0 and x = -2/5. 

 

Case 3:

What about quadratic equations in standard form?

For these types of quadratics, an algebraic technique involves factoring the quadratic into the product of two binomials. Rewriting a quadratic into factored form relies on using binomial identities. Here’s the first binomial identity to study.

Let’s look at an example.

Notice that a quadratic written as a binomial squared has only one solution. Look for the pattern of numbers in the standard form to be able to factor such quadratics into factored form.

Here’s another example.

This quadratic equation also has only one solution.

Now let’s look at quadratics that can factored into two binomials. For these, use this binomial identity.

A quadratic will factor into two binomials (x + a) and (x + b) if the terms a and b have the behavior shown in the identity. Let’s look at an example.

Here is a quadratic equation that is not a perfect square.

If this quadratic will factor into a product of two binomials, look at the constant term, 12.

Look at the factors of 12 that add up to the coefficient of the linear term, 7.

In other words, find two numbers, a and b, whose product is 12 and whose sum is 7.

Notice that the factors 3 and 4 have a product of 12 and a sum of 7. We can now solve the equation.

Here is another example.

Look at the factors of 15 that add up to the coefficient of the linear term, -8.

In other words, find two numbers, a and b, whose product is 15 and whose sum is -8.

Notice that the factors -3 and -5 have a product of 15 and a sum of -8. We can now solve the equation.