# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/quadratic-function/quadratic-equation-imaginary-root
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---

export const metadata = {
  title: "Imaginary or Non-Real Roots",
  description: "Learn when quadratic equations have imaginary roots, how to identify them using discriminant, and solve complex numbers step-by-step with examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/19/2025",
  subject: "Quadratic Functions",
};

## What are Non-Real Roots?

Quadratic equations <InlineMath math="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 <InlineMath math="i" />, where <InlineMath math="i = \sqrt{-1}" />. This means <InlineMath math="i^2 = -1" />.

Examples of imaginary numbers:

- <InlineMath math="3i" /> (read as: "three i")
- <InlineMath math="2 + 5i" /> (read as: "two plus five i")
- <InlineMath math="-4i" /> (read as: "negative four i")

Numbers like <InlineMath math="2 + 5i" /> are called complex numbers, because they are a combination of a real number <InlineMath math="2" /> and an imaginary number <InlineMath math="5i" />.

## When Does a Quadratic Equation Have Imaginary Roots?

A quadratic equation has imaginary roots when its discriminant is negative. The discriminant is <InlineMath math="D = b^2 - 4ac" />.

If <InlineMath math="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:

<div className="flex flex-col gap-4">
  <BlockMath math="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" />

  <BlockMath math="x = \frac{-b \pm \sqrt{-(4ac - b^2)}}{2a}" />
  <BlockMath math="x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}" />
</div>

### Example Problem 1

Let's find the roots of the equation <InlineMath math="x^2 + 4x + 5 = 0" />.

**Step 1**: Identify the values of <InlineMath math="a" />, <InlineMath math="b" />, and <InlineMath math="c" />.

- <InlineMath math="a = 1" />
- <InlineMath math="b = 4" />
- <InlineMath math="c = 5" />

**Step 2**: Calculate the discriminant.

<BlockMath math="D = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4" />

Since <InlineMath math="D = -4 < 0" />, this equation has imaginary roots.

**Step 3**: Use the quadratic formula.

<div className="flex flex-col gap-4">
  <BlockMath math="x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-4}}{2 \cdot 1}" />
  <BlockMath math="x = \frac{-4 \pm 2i}{2} = -2 \pm i" />
</div>

Therefore, the roots of the equation <InlineMath math="x^2 + 4x + 5 = 0" /> are <InlineMath math="x_1 = -2 + i" /> and <InlineMath math="x_2 = -2 - i" />.

### Example Problem 2

Determine the type of roots for the equation <InlineMath math="2x^2 + x + 3 = 0" />.

**Step 1**: Identify the values of <InlineMath math="a" />, <InlineMath math="b" />, and <InlineMath math="c" />.

- <InlineMath math="a = 2" />
- <InlineMath math="b = 1" />
- <InlineMath math="c = 3" />

**Step 2**: Calculate the discriminant.

<BlockMath math="D = b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot 3 = 1 - 24 = -23" />

Since <InlineMath math="D = -23 < 0" />, this equation has imaginary roots.

**Step 3**: Find the equation's roots.

<div className="flex flex-col gap-4">
  <BlockMath math="x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{-23}}{2 \cdot 2}" />
  <BlockMath math="x = \frac{-1 \pm i\sqrt{23}}{4}" />
</div>

Therefore, the roots of the equation are <InlineMath math="x_1 = \frac{-1 + i\sqrt{23}}{4}" /> and <InlineMath math="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 <InlineMath math="a + bi" /> and <InlineMath math="a - bi" />. These pairs are called "complex conjugates."

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