Write $(x - y)^2$ as an expression involving $x^2$, $y^2$, and $xy$. Remember the pattern for $(a - b)^2$.

Once you have $(x - y)^2$ expanded, replace $x^2 + y^2$ with the value given in the problem, then solve the resulting equation for $xy$.

After substitution, you will get a simple equation of the form $9 = 85 - 2xy$. Rearrange it to isolate $xy$.

In Desmos, enter the two equations:

• x^2 + y^2 = 85
• x - y = 3

Desmos will show where the circle and the line intersect.

Click on each intersection point of the circle and line. Desmos will display the coordinates $(x, y)$ of those points. Note the $x$ and $y$ values for either intersection (they will give the same product $xy$).

In a new expression line, type the product of the $x$- and $y$-coordinates you observed (for example, if one intersection is $(a, b)$, type a*b). The resulting value is the product $xy$ asked for in the problem.

You are given $x - y = 3$. Square both sides to connect this with $x^2 + y^2$:

Now you have an expression involving $x$ and $y$ that can be expanded and combined with $x^2 + y^2 = 85$.

Use the algebra identity

From step 1, we know $(x - y)^2 = 9$, so we can write

This equation links $x^2 + y^2$ and $xy$.

From the system, you know

Notice that $x^2 - 2xy + y^2$ can be rewritten as $(x^2 + y^2) - 2xy$. Substitute $x^2 + y^2 = 85$ into the equation from step 2:

Now you have a simple linear equation in terms of $xy$.

From

subtract 85 from both sides:

Divide both sides by $-2$:

So, the value of $xy$ is 38.