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Quadratic Equations with No Real Solution

Quadratic Equations with No Real Solution

Author: Sophia Tutorial

This lesson covers quadratic equations with no real solution.

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  • The Discriminant of the Quadratic Formula
  • Negative Square Roots and the Imaginary Unit
  • Complex Solutions to Quadratic Equations

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:

x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

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.  

i equals square root of negative 1 end root

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.

Terms to Know
Complex Number

a number containing a real component and an imaginary component

Imaginary Number

a non-real number which is a multiple of i, sqrt(-1)

Formulas to Know
Imaginary Number

i equals square root of negative 1 end root