$$ (3+\sqrt{5}) $$ and $$ (3-\sqrt{5}) $$

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The expressions $$ (3 + \sqrt{5}) $$ and $$ (3 - \sqrt{5}) $$ are **conjugate pairs** involving the square root of 5. These numbers have several interesting mathematical properties and applications. Here's a detailed overview:

### 1. **Basic Properties**

- **Sum and Difference:**
  $$
  \begin{align*}
  \text{Sum} &= (3 + \sqrt{5}) + (3 - \sqrt{5}) = 6 \
  \text{Difference} &= (3 + \sqrt{5}) - (3 - \sqrt{5}) = 2\sqrt{5}
  \end{align*}
  $$

- **Product:**
  $$
  (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4
  $$

- **Quotient:**
  $$
  \frac{3 + \sqrt{5}}{3 - \sqrt{5}} = \frac{(3 + \sqrt{5})^2}{(3)^2 - (\sqrt{5})^2} = \frac{9 + 6\sqrt{5} + 5}{4} = \frac{14 + 6\sqrt{5}}{4} = \frac{7 + 3\sqrt{5}}{2}
  $$

### 2. **Roots of a Quadratic Equation**

These two numbers are the roots of the quadratic equation derived from their sum and product.

- **Quadratic Equation:**
  $$
  x^2 - (\text{Sum})x + (\text{Product}) = 0 \implies x^2 - 6x + 4 = 0
  $$

Solving $$ x^2 - 6x + 4 = 0 $$ using the quadratic formula:
  $$
  x = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm \sqrt{20}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5}
  $$

### 3. **Irrationality**

Both $$ 3 + \sqrt{5} $$ and $$ 3 - \sqrt{5} $$ are **irrational numbers** since they involve the square root of a non-square integer (5).

### 4. **Recurrence Relations and Powers**

These numbers can be used to express terms in certain recurrence relations, especially those resembling the Fibonacci sequence.

- **Powers:**
  The powers of $$ (3 + \sqrt{5}) $$ and $$ (3 - \sqrt{5}) $$ can be expressed using binomial expansion. Additionally, since they are roots of the quadratic equation $$ x^2 = 6x - 4 $$, higher powers can be simplified using this relation.

- **Example:**
  $$
  (3 + \sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5} = 6(3 + \sqrt{5}) - 4
  $$
  This simplifies calculations involving higher powers.

### 5. **Geometric Interpretation**

In the context of the quadratic equation $$ x^2 - 6x + 4 = 0 $$, the numbers $$ 3 + \sqrt{5} $$ and $$ 3 - \sqrt{5} $$ represent the points where the parabola $$ y = x^2 - 6x + 4 $$ intersects the x-axis.

### 6. **Applications**

- **Algebraic Identities:**
  They are useful in simplifying expressions involving radicals.
  
- **Number Theory:**
  These conjugate pairs appear in the study of quadratic fields and Diophantine equations.

- **Geometry:**
  They can represent lengths, distances, or other geometric quantities that involve irrational numbers.

### 7. **Graphical Representation**

If you plot these numbers on the real number line:

- $$ 3 + \sqrt{5} $$ is approximately $$ 3 + 2.236 = 5.236 $$
- $$ 3 - \sqrt{5} $$ is approximately $$ 3 - 2.236 = 0.764 $$

They are symmetric around the center point 3.

### 8. **Conjugate Properties in Complex Numbers**

While $$ (3 + \sqrt{5}) $$ and $$ (3 - \sqrt{5}) $$ are real numbers, the concept of conjugates extends to complex numbers, where conjugate pairs are of the form $$ a + bi $$ and $$ a - bi $$. These pairs share similar properties, such as having the same real part and opposite imaginary parts.

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If you have a specific question or application in mind regarding $$ (3 + \sqrt{5}) $$ and $$ (3 - \sqrt{5}) $$, feel free to ask for a more targeted explanation!
$$ (3 + \sqrt{5}) $$ and $$ (3 - \sqrt{5}) $$ are conjugate pairs of irrational numbers. They are roots of the equation $$ x^2 - 6x + 4 = 0 $$ and have properties like sum of 6, product of 4, and can be used in algebraic expressions and geometric interpretations.

\( (3+\sqrt{5}) \) and \( (3-\sqrt{5}) \)

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