Table of Contents |
Working with the quadratic formula is one method in determining if there are no real solutions to a quadratic equation. Recall the quadratic formula:
The expression that is underneath the square root is called the discriminant. Because the discriminant is underneath a square root sign, it must not have a negative value, otherwise it does not evaluate to a real number. This is how we can tell if a quadratic has no real solutions by using the quadratic formula.
EXAMPLE
Find the solutions for the quadratic equation .Identify the values for b, and c in the equation | |
Substitute these values in the quadratic formula | |
In the discriminant, square -5 and multiply 4, 1, and 8 | |
No real solutions |
Even though some quadratic equations may have no real solutions, we can still express their solutions mathematically. To do so, we use the imaginary number, i, in the expression for its solution. The imaginary number, i, is a non-real number that represents the square root of -1.
The letter i is used to denote the square root of negative 1. We can rewrite the square roots of negative numbers using this letter.
EXAMPLE
If we encounter a negative value underneath the radical when using the quadratic formula, we can express the solutions to the quadratic equation using complex numbers. A complex number contains a real part and an imaginary part, such as or .
EXAMPLE
Suppose we were calculating the solutions to a quadratic equation and got to this step:Rewrite square root | |
Evaluate the square root of 16 and -1 | |
Create two separate solutions, one addition and one subtraction | |
Divide each term by 2 | |
Our solutions |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License