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

It looks like you're working with the expressions \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \). These two expressions are **conjugates** of each other. Conjugate pairs like these have several interesting properties, especially in the context of algebra and quadratic equations. Here's a breakdown of their key properties and how they relate to each other: ### 1. **Sum and Product** - **Sum:** \[ (3 + \sqrt{5}) + (3 - \sqrt{5}) = 3 + \sqrt{5} + 3 - \sqrt{5} = 6 \] - **Product:** \[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \] These results are derived from the algebraic identity for the product of conjugates: \[ (a + b)(a - b) = a^2 - b^2 \] ### 2. **Quadratic Equation** Since \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) are roots of a quadratic equation, you can construct that equation using their sum and product. For a quadratic equation of the form: \[ x^2 - ( \text{sum of roots} )x + ( \text{product of roots} ) = 0 \] Plugging in the sum and product: \[ x^2 - 6x + 4 = 0 \] So, the quadratic equation with roots \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) is: \[ x^2 - 6x + 4 = 0 \] ### 3. **Other Properties** - **Irrationality:** Both \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) are irrational numbers because they involve the square root of a non-square integer (5). - **Reciprocals:** \[ \frac{1}{3 + \sqrt{5}} = \frac{3 - \sqrt{5}}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{3 - \sqrt{5}}{4} \] Similarly, \[ \frac{1}{3 - \sqrt{5}} = \frac{3 + \sqrt{5}}{4} \] - **Powers:** Raising these expressions to higher powers can be simplified using their quadratic relationship. For example: - **Square:** \[ (3 + \sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5} \] \[ (3 - \sqrt{5})^2 = 9 - 6\sqrt{5} + 5 = 14 - 6\sqrt{5} \] - **Recurrence Relations:** Due to their role as roots of a quadratic equation, sequences involving these terms can often be expressed using linear recurrence relations. ### 4. **Applications** Conjugate pairs like \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) frequently appear in various areas of mathematics, including: - **Solving Quadratic Equations:** As shown, they serve as solutions to specific quadratic equations. - **Number Theory:** They can be involved in problems related to algebraic integers. - **Geometry:** Expressions involving square roots are common in geometric calculations, such as distance formulas. - **Continued Fractions:** Conjugate irrationals can be used to study properties of continued fractions. ### 5. **Graphical Representation** If you were to plot these as points on the real number line: - Both points are located at a distance of \( \sqrt{5} \) from the point 3. - \( (3 + \sqrt{5}) \) is to the right of 3, while \( (3 - \sqrt{5}) \) is to the left. ### Summary - **Conjugates:** \( (3 + \sqrt{5}) \) and \( (3 - \sqrt{5}) \) are a pair of conjugate expressions. - **Sum:** 6 - **Product:** 4 - **Quadratic Equation:** \( x^2 - 6x + 4 = 0 \) - **Properties:** Irrational, have reciprocal relationships, and can be manipulated using algebraic identities. If you have a specific question or need further elaboration on a particular aspect of these expressions, feel free to ask!