The Discriminant in the Quadratic Formula
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. This is illustrated in the example below:
Because we have a negative number underneath the square root, we can conclude that this equation has no real solutions.
Negative Square Roots and the Imaginary Unit
Even though some quadratic equations may have no real solutions, we can still express their solutions mathematically. To do so, we use the imaginary unit, i, in the expression for its solution.
The letter i is used to denote the square root of negative 1. We can rewrite square roots of negative number using this letter. This is shown in the examples below:
Complex Solutions to Quadratic Equations
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.
In our previous example, the complex number 5 ± 4i was being divided by 2. We can divide 5 and 4i by 2 separately to arrive at 2.5 ± 2i. Then, we can create two expressions, one taking the minus sign, and the other taking the plus sign, due to the ± symbol.