Problem Objective

The objective is to calculate the value of the expression $x^2 + y^2$.

Given Values

We are provided with the values of $x$ and $y$:

• $x = 3 + \sqrt{5}$
• $y = 3 - \sqrt{5}$

Solution Steps

To efficiently find $x^2 + y^2$, we can use algebraic identities. A suitable identity is $x^2 + y^2 = (x+y)^2 - 2xy$. We first calculate the sum ($x+y$) and the product ($xy$) of the given values.

Calculate Sum ($x+y$)

Add the expressions for $x$ and $y$:

$x+y = (3 + \sqrt{5}) + (3 - \sqrt{5})$

Combine like terms:

$x+y = 3 + 3 + \sqrt{5} - \sqrt{5}$

$x+y = 6$

Calculate Product ($xy$)

Multiply the expressions for $x$ and $y$. This utilizes the difference of squares formula $(a+b)(a-b) = a^2 - b^2$:

$xy = (3 + \sqrt{5})(3 - \sqrt{5})$

Apply the formula:

$xy = 3^2 - (\sqrt{5})^2$

$xy = 9 - 5$

$xy = 4$

Apply Identity for $x^2 + y^2$

Substitute the calculated values of $x+y = 6$ and $xy = 4$ into the identity $x^2 + y^2 = (x+y)^2 - 2xy$:

$x^2 + y^2 = (6)^2 - 2(4)$

Perform the calculations:

$x^2 + y^2 = 36 - 8$

$x^2 + y^2 = 28$

Final Answer

The value of $x^2 + y^2$ is 28.