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To solve part (a) of the problem, we need to perform a change of variables. The function given is:

\[
f(x, y) = 3x^2 + 4xy + 3y^2
\]

We are asked to change variables using the transformations \( u = x + y \) and \( v = x - y \), and then rewrite the function \( f(x, y) \) in terms of \( u \) and \( v \).

### Step 1: Express \( x \) and \( y \) in terms of \( u \) and \( v \).

Given:
\[
u = x + y
\]
\[
v = x - y
\]

To find \( x \) and \( y \) in terms of \( u \) and \( v \):

Add the two equations:
\[
u + v = (x + y) + (x - y) = 2x
\]
Thus,
\[
x = \frac{u + v}{2}
\]

Subtract the second equation from the first:
\[
u - v = (x + y) - (x - y) = 2y
\]
Thus,
\[
y = \frac{u - v}{2}
\]

### Step 2: Substitute \( x = \frac{u + v}{2} \) and \( y = \frac{u - v}{2} \) into the original function.

Now, substitute these expressions into \( f(x, y) \):

\[
f(x, y) = 3x^2 + 4xy + 3y^2
\]

Substituting \( x \) and \( y \):

\[
f\left(\frac{u + v}{2}, \frac{u - v}{2}\right) = 3\left(\frac{u + v}{2}\right)^2 + 4\left(\frac{u + v}{2}\right)\left(\frac{u - v}{2}\right) + 3\left(\frac{u - v}{2}\right)^2
\]

### Step 3: Simplify the expression.

Calculate each term separately:

1. \( 3\left(\frac{u + v}{2}\right)^2 \):
   \[
   3\left(\frac{u + v}{2}\right)^2 = 3 \times \frac{(u + v)^2}{4} = \frac{3(u^2 + 2uv + v^2)}{4}
   \]

2. \( 4\left(\frac{u + v}{2}\right)\left(\frac{u - v}{2}\right) \):
   \[
   4\left(\frac{u + v}{2}\right)\left(\frac{u - v}{2}\right) = 4 \times \frac{(u + v)(u - v)}{4} = (u^2 - v^2)
   \]

3. \( 3\left(\frac{u - v}{2}\right)^2 \):
   \[
   3\left(\frac{u - v}{2}\right)^2 = 3 \times \frac{(u - v)^2}{4} = \frac{3(u^2 - 2uv + v^2)}{4}
   \]

Adding these terms together:

\[
f(u, v) = \frac{3(u^2 + 2uv + v^2)}{4} + (u^2 - v^2) + \frac{3(u^2 - 2uv + v^2)}{4}
\]

Simplify the expression:

Combine like terms:

\[
f(u, v) = \left(\frac{3}{4}u^2 + \frac{3}{4}v^2 + \frac{3}{2}uv\right) + (u^2 - v^2) + \left(\frac{3}{4}u^2 + \frac{3}{4}v^2 - \frac{3}{2}uv\right)
\]

Notice that the cross terms \( \frac{3}{2}uv \) cancel out:

\[
f(u, v) = \frac{3}{2}u^2 + \frac{3}{2}v^2 + u^2 - v^2
\]

\[
f(u, v) = \left(\frac{3}{2} + 1\right)u^2 + \left(\frac{3}{2} - 1\right)v^2
\]

\[
f(u, v) = \frac{5}{2}u^2 + \frac{1}{2}v^2
\]

Thus, the new function in terms of \( u \) and \( v \) is:

\[
f(u, v) = \frac{5}{2}u^2 + \frac{1}{2}v^2
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do the level curves of the function \( f(u, v) = \frac{5}{2}u^2 + \frac{1}{2}v^2 \) compare to those of the original function \( f(x, y) = 3x^2 + 4xy + 3y^2 \)?
2. What is the geometric interpretation of the transformation \( u = x + y \) and \( v = x - y \)?
3. How would you sketch the graph of the function \( f(u, v) = \frac{5}{2}u^2 + \frac{1}{2}v^2 \)?
4. How do the eigenvalues of the original quadratic form \( 3x^2 + 4xy + 3y^2 \) relate to the coefficients in the transformed function?
5. What other transformations could simplify a quadratic form like this one?

**Tip:** When simplifying quadratic forms, completing the square can often help in recognizing patterns or making the problem easier to work with.

Learn how to transform the quadratic function f(x, y) = 3x^2 + 4xy + 3y^2 into a function in terms of new variables u and v using a change of variables technique. This detailed solution covers completing the square, algebraic transformations, and the geometric interpretation of variable transformations.

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