A Word about Quadratic Formula Solutions!

Let's discuss irrational results achieved when using the Quadratic Formula. In algebra, all radicals must be simplified and all fractions must be reduced to lowest terms.

We usually do not change the solutions to decimal approximations. There is one exception to this rule which is when we graph functions.

Simplifying Radicals:

A square root that is an irrational number is simplified if the radicand has no factors raised to a power greater than or equal to the index.

Example 1:

Simplify .

We know that 24 = 4(6) or 2²(6).

Then .

Example 2:

Simplify .

We know that 75 = 25(3) or 5²(5).

Then .

Example 3:

Simplify .

We know that 18 = 9(2) or 3²(2).

Then .

 

Simplifying and Reducing Quadratic Formula Solutions:

Example 4:

Given ,simplify and reduce to lowest terms.

First, we simplify the radical.

Now we notice that ALL terms have the factor of 2 in common. We will first factor it out of the numerator as follows:

Finally, we will reduce as follows:

 

Use a Calculator to change a Quadratic Formula Solution to a Decimal Number:

Example 5:

Change to a decimal approximation.

Here are the calculator steps to find the most exact decimal approximatation. NOTE: The entire numerator MUST be enclosed in parentheses.

We find that is approximately equal to 0.224621125 ... . This number has infinitely many decimal places. It is called an irrational number.