Solving the Equation x^2 - y^2 for Given Values of x and y

In this article, we will explore a problem involving the algebraic manipulation of given equations to solve for a specific expression. Specifically, we will calculate the value of x^2 - y^2 where x (√5 1) / (√5 - 1) and y (√5 - 1) / (√5 1). This problem involves basic algebraic operations and can be solved step-by-step. Let's break it down and explore the solution in detail.

Given Values and Expressions

We are provided with the following expressions:

The objective is to find the value of (x^2 - y^2).

Step-by-Step Solution

Step 1: Simplify the Expressions for x and y

Let's start by simplifying the expressions for x and y.

x (frac{sqrt{5} 1}{sqrt{5} - 1})

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is (sqrt{5} 1):

x (frac{sqrt{5} 1 cdot (sqrt{5} 1)}{(sqrt{5} - 1) cdot (sqrt{5} 1)} frac{5 2sqrt{5} 1}{5 - 1} frac{6 2sqrt{5}}{4} frac{3 sqrt{5}}{2})

y (frac{sqrt{5} - 1}{sqrt{5} 1})

Similarly, rationalize the denominator by multiplying both the numerator and the denominator by the conjugate (sqrt{5} - 1):

y (frac{sqrt{5} - 1 cdot (sqrt{5} - 1)}{(sqrt{5} 1) cdot (sqrt{5} - 1)} frac{5 - 2sqrt{5} 1}{5 - 1} frac{6 - 2sqrt{5}}{4} frac{3 - sqrt{5}}{2})

Now that we have simplified x and y, let's proceed to the next step.

Step 2: Substitute x and y into the Expression x^2 - y^2

To find the value of (x^2 - y^2), we can use the identity (a^2 - b^2 (a b)(a - b)).

Let (a x) and (b y), then:

(x^2 - y^2 (x y)(x - y))

Now calculate x y and x - y:

x y frac{3 sqrt{5}}{2} frac{3 - sqrt{5}}{2} frac{6}{2} 3

x - y frac{3 sqrt{5}}{2} - frac{3 - sqrt{5}}{2} frac{2sqrt{5}}{2} sqrt{5}

Therefore, (x^2 - y^2 (x y)(x - y) 3sqrt{5})

Conclusion

The value of (x^2 - y^2) is (3sqrt{5}). This exercise demonstrates the importance of algebraic manipulation and the use of identities in solving complex expressions.

Keywords: Solving Equations, Algebraic Manipulation, Mathematical Operations