The aim of this question is to **qualitatively evaluate the roots of a polynomial** using prior knowledge of algebra.

As an example, let’s **consider a standard quadratic equation**:

\[ a x^{ 2 } \ + \ b x \ + \ c \ = \ 0 \]

The **roots of such a quadric equation** are given by:

\[ \lambda_{1,2} \ = \ \dfrac{ -b \ \pm \ \sqrt{ b^{ 2 } \ – \ 4 a c } }{ 2 a } \]

Here, one may notice that the **two roots are conjugates of each other**.

A **conjugate pair** of roots is the one where two roots have the **same non-square root term** but their **s****quare root terms are equal and opposite** in sign.

## Expert Answer

Given that:

\[ \lambda_1 \ = \ 2 \ + \ \sqrt{ 3 } \]

If we **assume that the polynomial has a degree of 2**:

\[ a x^{ 2 } \ + \ b x \ + \ c \ = \ 0 \]

Then we know that the **roots of such a quadric equation** are given by:

\[ \lambda_{1,2} \ = \ \dfrac{ -b \ \pm \ \sqrt{ b^{ 2 } \ – \ 4 a c } }{ 2 a } \]

This shows that the **two roots** $ \lambda_1 $ and $ \lambda_2 $ are **conjugates of each other**. So if $ 2 \ + \ \sqrt{ 3 } $ is one root then $ 2 \ – \ \sqrt{ 3 } $ must be the other root.

Here, we have assumed that the equation is quadratic. However, **this fact is true for any polynomial of order higher than two**.

## Numerical Result

If $ 2 \ + \ \sqrt{ 3 } $ is one root, then $ 2 \ – \ \sqrt{ 3 } $ must be the other root.

## Example

Given the equation $ x^{ 2 } \ + \ 2 x \ + \ 4 \ = \ 0 $, **find its roots**.

Comparing the given equation with the following **standard quadratic equation**:

\[ a x^{ 2 } \ + \ b x \ + \ c \ = \ 0 \]

We can see that:

\[ a \ = \ 1, \ b \ = \ 2 \text{ and } \ c \ = \ 4 \]

**Roots of such a quadric equation** are given by:

\[ \lambda_{1,2} \ = \ \dfrac{ -b \ \pm \ \sqrt{ b^{ 2 } \ – \ 4 a c } }{ 2 a } \]

**Substituting values:**

\[ \lambda_{1,2} \ = \ \dfrac{ -2 \ \pm \ \sqrt{ 2^{ 2 } \ – \ 4 ( 1 ) ( 4 ) } }{ 2 ( 1 ) } \]

\[ \lambda_{1,2} \ = \ \dfrac{ -2 \ \pm \ \sqrt{ 4 \ – \ 16 } }{ 2 } \]

\[ \lambda_{1,2} \ = \ \dfrac{ -2 \ \pm \ \sqrt{ -12 } }{ 2 } \]

\[ \lambda_{1,2} \ = \ -1 \ \pm \ \sqrt{ -3 } \]

\[ \lambda_{1,2} \ = \ -1 \ \pm \ \sqrt{ 3 } i \]

**Which are the roots of the given equation.**