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

It looks like you've provided the two expressions \( (3+\sqrt{5}) \) and \( (3-\sqrt{5}) \). These are known as **conjugate pairs** because they differ only in the sign between their terms. Conjugate pairs often arise in various mathematical contexts, such as solving quadratic equations, simplifying expressions, and working with complex numbers. Here's a detailed exploration of their properties and significance: ### 1. **Sum and Product** - **Sum**: \[ (3 + \sqrt{5}) + (3 - \sqrt{5}) = 3 + 3 + \sqrt{5} - \sqrt{5} = 6 \] - **Product**: \[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \] This uses the **difference of squares** formula: \( (a + b)(a - b) = a^2 - b^2 \). ### 2. **Quadratic Equation Roots** Given that \( (3+\sqrt{5}) \) and \( (3-\sqrt{5}) \) are roots, we can form a quadratic equation using Vieta's formulas, where the sum of the roots is \( 6 \) and the product is \( 4 \): \[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \] \[ x^2 - 6x + 4 = 0 \] So, \( 3+\sqrt{5} \) and \( 3-\sqrt{5} \) are the solutions to the equation \( x^2 - 6x + 4 = 0 \). ### 3. **Reciprocals** The reciprocal of each expression can be simplified as follows: - **Reciprocal of \( (3 + \sqrt{5}) \)**: \[ \frac{1}{3 + \sqrt{5}} = \frac{3 - \sqrt{5}}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{3 - \sqrt{5}}{4} \] - **Reciprocal of \( (3 - \sqrt{5}) \)**: \[ \frac{1}{3 - \sqrt{5}} = \frac{3 + \sqrt{5}}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{3 + \sqrt{5}}{4} \] ### 4. **Estimation** To get a numerical sense of these expressions: - \( \sqrt{5} \approx 2.236 \) - Therefore: \[ 3 + \sqrt{5} \approx 3 + 2.236 = 5.236 \] \[ 3 - \sqrt{5} \approx 3 - 2.236 = 0.764 \] ### 5. **Irrationality** Both \( (3+\sqrt{5}) \) and \( (3-\sqrt{5}) \) are **irrational numbers** since they involve \( \sqrt{5} \), which is itself irrational. This means they cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions. ### 6. **Applications** - **Algebraic Manipulations**: Conjugate pairs are used to rationalize denominators and simplify expressions involving radicals. - **Quadratic Equations**: As shown, they serve as roots to specific quadratic equations. - **Recurrence Relations**: In sequences defined by second-order linear recurrence relations, such as the Fibonacci sequence, expressions involving conjugate pairs can describe closed-form solutions. - **Complex Numbers**: While your expressions are real, the concept of conjugates extends to complex numbers, where they play a crucial role in simplifying expressions and solving equations. ### 7. **Graphical Representation** If you were to plot these numbers on a number line: - \( 3 + \sqrt{5} \approx 5.236 \) would be to the right of \( 3 \). - \( 3 - \sqrt{5} \approx 0.764 \) would be to the left of \( 3 \). They are symmetrically placed around the point \( 3 \) on the number line, separated by \( \sqrt{5} \). ### 8. **Additional Properties** - **Multiplicative Inverses**: As shown in the reciprocals section, they are inverses scaled by their product. - **Exponents and Powers**: Raising these expressions to powers will involve binomial expansions and can result in expressions with and without radicals, depending on the power. ### 9. **Practical Example** Suppose you're solving a problem where you're given that the sum and product of two numbers are \( 6 \) and \( 4 \) respectively. The numbers must be \( 3+\sqrt{5} \) and \( 3-\sqrt{5} \). --- If you had a specific question or application in mind regarding these expressions, please feel free to provide more details!