Complex & Irrational Roots: Definitions & Examples

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson defines complex roots, irrational roots and quadratic equations. Examples of how to solve quadratic equations with complex and irrational solutions are provided.

Quadratic Equations

Suppose you are building a square pen in your back yard for your dog and you want your dog to have an area of four meters to play in. How long would the lengths of the sides need to be? We can model this situation with a quadratic equation. A quadratic equation is an equation where the highest exponent is two.

If we use x to represent the length of a side of the pen, we can write an equation to find the length. We know that the area needs to be 4 meters and the area of a square is the side squared.


In order to solve this equation, we need to take the square root of each side. When we take the square root of 4, we get 2 or -2. In this instance, -2 could not be an answer because we can't measure - 2 meters. Therefore, each side of the pen needs to be two meters in length.

We can graph the equation we set up in the example above on the coordinate plane. To graph the equation, we subtract 4 from both sides and set the equation equal to y.



The solutions to the equation were 2 and -2. If you look at the graph above, notice that the graph intersects the x-axis at 2 and -2. The solutions to the quadratic equation are the same as the x-intercepts of the graph. These solutions and x-intercepts are also called the roots of the equation. Notice they are whole numbers, and we can see them easily on the graph.

Irrational Roots

What happens if the graph of a quadratic equation doesn't have integer x-intercepts?



The graph above does not cross the x-axis at integer points. We know that it crosses somewhere between -1 and -2, and 1 and 2. To find exactly where this graph crosses, we need to look at the equation. We want to know the x-intercepts, so really we need to know what x is when y = 0. So we will replace y with 0. Then we need to solve for x, so we add three to both sides.


Now to solve for x, we must take the square root of each side.

square roots

Since 3 is not a perfect square, the square root is an irrational number. An irrational number is a number that cannot be written as a fraction, a/b, where a and b are integers. It is a decimal that does not repeat or end. The square root of 3 is an irrational number. However, we can take the square root and round the decimal. This means that we will get an approximate answer.


We can check our solutions by looking back at the graph. Recall that we said the solutions were between -2, and -1 and between 1 and 2. Our solutions of -1.7 and 1.7 are between those two values.

Complex Roots

So far, we've seen where the solutions could be integer points on the graph, or where they are irrational. There is another possible solution of a quadratic equation, and that is if the graph doesn't cross the x-axis at all.

complex graph

Notice the graph does not cross the x-axis. This means that there are no real solutions. However, we do have complex solutions. Complex solutions or roots are numbers that have an imaginary part to them. The imaginary part, i, is found when taking the square root of a negative number.

The equation for the graph above is:


If we wanted to solve this equation, we would follow the same procedures as above. First, we will replace y with 0. Then we will subtract 4 from both sides.


Now we need to take the square root of both sides.

square roots

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