Imaginary or Non-Real Roots

What are Non-Real Roots?

Quadratic equations $ax^2 + bx + c = 0$ sometimes have solutions that cannot be found in ordinary numbers. These solutions are called "non-real roots" or "imaginary roots."

Imagine we're looking for a number that, when multiplied by itself, gives a negative result. Does such a number exist? No! Because any number multiplied by itself always gives a positive result or zero. This is where the concept of imaginary numbers begins.

Imaginary Numbers

Imaginary numbers are numbers that contain $i$, where $i = \sqrt{-1}$. This means $i^2 = -1$.

Examples of imaginary numbers:

• $3i$ (read as: "three i")
• $2 + 5i$ (read as: "two plus five i")
• $-4i$ (read as: "negative four i")

Numbers like $2 + 5i$ are called complex numbers, because they are a combination of a real number $2$ and an imaginary number $5i$.

When Does a Quadratic Equation Have Imaginary Roots?

A quadratic equation has imaginary roots when its discriminant is negative. The discriminant is $D = b^2 - 4ac$.

If $D < 0$, then the quadratic equation will have two different imaginary roots.

How to Find Imaginary Roots

To find imaginary roots, we still use the formula:

Example Problem 1

Let's find the roots of the equation $x^2 + 4x + 5 = 0$.

Step 1: Identify the values of $a$, $b$, and $c$.

• $a = 1$
• $b = 4$
• $c = 5$

Step 2: Calculate the discriminant.

Since $D = -4 < 0$, this equation has imaginary roots.

Step 3: Use the quadratic formula.

Therefore, the roots of the equation $x^2 + 4x + 5 = 0$ are $x_1 = -2 + i$ and $x_2 = -2 - i$.

Example Problem 2

Determine the type of roots for the equation $2x^2 + x + 3 = 0$.

Step 1: Identify the values of $a$, $b$, and $c$.

• $a = 2$
• $b = 1$
• $c = 3$

Step 2: Calculate the discriminant.

Since $D = -23 < 0$, this equation has imaginary roots.

Step 3: Find the equation's roots.

Therefore, the roots of the equation are $x_1 = \frac{-1 + i\sqrt{23}}{4}$ and $x_2 = \frac{-1 - i\sqrt{23}}{4}$.

Why Do Imaginary Roots Always Come in Pairs?

Imaginary roots always appear in pairs in the form of $a + bi$ and $a - bi$. These pairs are called "complex conjugates."

This happens because the quadratic formula involves $\pm\sqrt{D}$. When $D < 0$, we get $\pm i\sqrt{|D|}$, which gives us complex conjugate pairs.