If the sum of the square roots of two integers is sqrt(9+6sqrt(2)), what is the sum of the squares of...

1. Translate the problem requirements: We need to find two integers whose square roots sum to \(\sqrt{9+6\sqrt{2}}\), then calculate the sum of the squares of those original integers.
2. Recognize the nested radical pattern: The expression \(\sqrt{9+6\sqrt{2}}\) suggests it can be simplified to a sum of simpler square roots of the form \(\sqrt{a} + \sqrt{b}\).
3. Set up the algebraic relationship: If \(\sqrt{a} + \sqrt{b} = \sqrt{9+6\sqrt{2}}\), then squaring both sides will help us identify the values of a and b.
4. Solve for the individual integers: Use the squared equation to find the specific values of a and b that satisfy our conditions.
5. Calculate the final sum: Add the squares of the two integers we found to get our answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We have two integers - let's call them a and b for now. The problem tells us that when we take the square root of each integer and add them together, we get \(\sqrt{9+6\sqrt{2}}\).

So we can write this as: \(\sqrt{a} + \sqrt{b} = \sqrt{9+6\sqrt{2}}\)

Our goal is to find what \(a + b\) equals (the sum of the squares of these integers, since a and b are the integers themselves).

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship

2. Recognize the nested radical pattern

Now, looking at \(\sqrt{9+6\sqrt{2}}\), this expression has a very specific pattern. It looks like it might be the result of squaring something simpler.

Think about it this way: if we had two simple square roots like \(\sqrt{p} + \sqrt{q}\), and we squared that expression, what would we get?

\((\sqrt{p} + \sqrt{q})^2 = p + 2\sqrt{pq} + q = (p + q) + 2\sqrt{pq}\)

Comparing this to our expression \(9 + 6\sqrt{2}\), we can see:

- The whole number part: \(p + q = 9\)

- The radical part: \(2\sqrt{pq} = 6\sqrt{2}\)

From the second equation: \(\sqrt{pq} = 3\sqrt{2}\), so \(pq = 18\)

3. Set up the algebraic relationship

Now we have two simple equations to work with:

- \(p + q = 9\)

- \(pq = 18\)

These are much easier to work with than the original nested radical! We need to find two numbers that add to 9 and multiply to 18.

Let's think through this systematically. What pairs of numbers multiply to 18?

- \(1 \times 18 = 18\), and \(1 + 18 = 19\) (too big)

- \(2 \times 9 = 18\), and \(2 + 9 = 11\) (too big)

- \(3 \times 6 = 18\), and \(3 + 6 = 9\) ✓

Perfect! So \(p = 3\) and \(q = 6\) (or vice versa).

Process Skill: SIMPLIFY - Breaking down the complex nested radical into manageable parts

4. Solve for the individual integers

Now we can verify our work. If \(p = 3\) and \(q = 6\), then:

\(\sqrt{9+6\sqrt{2}} = \sqrt{3} + \sqrt{6}\)

Let's check this by squaring the right side:

\((\sqrt{3} + \sqrt{6})^2 = 3 + 2\sqrt{3 \times 6} + 6 = 9 + 2\sqrt{18} = 9 + 2\sqrt{9 \times 2} = 9 + 2 \times 3\sqrt{2} = 9 + 6\sqrt{2}\) ✓

Excellent! This confirms that \(\sqrt{a} + \sqrt{b} = \sqrt{3} + \sqrt{6}\), which means our two integers are \(a = 3\) and \(b = 6\).

5. Calculate the final sum

Now we need to find the sum of the squares of these integers:

\(a^2 + b^2 = 3^2 + 6^2 = 9 + 36 = 45\)

Final Answer

The sum of the squares of the two integers is 45, which corresponds to answer choice (C).

To verify: We found that the two integers are 3 and 6, their square roots sum to \(\sqrt{9+6\sqrt{2}}\) as required, and \(3^2 + 6^2 = 9 + 36 = 45\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what the question is asking for

Students often confuse "the sum of the squares of these two integers" with "the squares of the sum of these integers" or think they need to find the actual integers themselves rather than the sum of their squares. The question asks for \(a^2 + b^2\) where a and b are the integers, not \((a + b)^2\).

2. Failing to recognize the nested radical pattern

Many students see \(\sqrt{9+6\sqrt{2}}\) and try to simplify it using calculator approximations or attempt algebraic manipulation without recognizing that this expression likely came from squaring a sum of square roots. They miss the key insight that expressions of the form \(\sqrt{p + q\sqrt{r}}\) often simplify to \(\sqrt{m} + \sqrt{n}\) for some integers m and n.

3. Setting up the wrong variable relationships

Students may correctly identify that \(\sqrt{a} + \sqrt{b} = \sqrt{9+6\sqrt{2}}\) but then make errors in the setup, such as assuming that a and b must equal 9 and 6 directly, rather than understanding that they need to "denest" the radical first by finding simpler square roots that sum to the given expression.

Errors while executing the approach

1. Algebraic errors when expanding \((\sqrt{p} + \sqrt{q})^2\)

When expanding \((\sqrt{p} + \sqrt{q})^2 = p + 2\sqrt{pq} + q\), students frequently forget the middle term \(2\sqrt{pq}\) or make sign errors. This is crucial because correctly identifying that \(2\sqrt{pq} = 6\sqrt{2}\) leads to the key equation \(pq = 18\).

2. Incorrect solving of the system \(p + q = 9\), \(pq = 18\)

Students may struggle to find factor pairs systematically or make arithmetic errors. Common mistakes include not checking all factor pairs of 18, or incorrectly calculating sums (for example, thinking \(2 + 9 = 10\) instead of 11).

3. Errors in verification step

When checking that \((\sqrt{3} + \sqrt{6})^2 = 9 + 6\sqrt{2}\), students often make mistakes in simplifying \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\), or they may skip the verification entirely, missing the chance to catch earlier errors.

Errors while selecting the answer

1. Confusing intermediate values with the final answer

Students might select 9 (which was \(p + q\)) or 18 (which was \(pq\)) instead of calculating \(3^2 + 6^2 = 45\). They may also select the sum of the integers (\(3 + 6 = 9\)) rather than the sum of their squares.

2. Computational errors in final calculation

Even after correctly identifying that the integers are 3 and 6, students may make simple arithmetic mistakes: calculating \(3^2 + 6^2\) as \(6 + 36 = 42\) (forgetting to square the 3) or \(9 + 6 = 15\) (forgetting to square the 6).