Solving for $x^2 + y^2$

We are given two equations:

• $x + y = 9$
• $xy = 18$

We need to find the value of $x^2 + y^2$. We can use the algebraic identity $(x+y)^2 = x^2 + 2xy + y^2$.

Using the Algebraic Identity

Rearranging the identity to solve for $x^2 + y^2$, we get:

$x^2 + y^2 = (x+y)^2 - 2xy$

Calculation

Now, substitute the given values of $x+y$ and $xy$ into the rearranged formula:

1. Substitute $x+y = 9$ and $xy = 18$:

   $x^2 + y^2 = (9)^2 - 2(18)$

2. Calculate the square of 9:

   $x^2 + y^2 = 81 - 2(18)$

3. Calculate the product of 2 and 18:

   $x^2 + y^2 = 81 - 36$

4. Perform the subtraction:

   $x^2 + y^2 = 45$

Therefore, the value of $x^2 + y^2$ is 45.