If x^2 + 1/x^2 = 4, what is the value of x^4 + 1/x^4 ?

1. Translate the problem requirements: We're given that \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2} = 4\), and we need to find the value of \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\). This means we need to connect these two expressions algebraically.
2. Establish the connection pattern: Recognize that \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\) can be expressed in terms of \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2}\) by treating \(\mathrm{x}^2\) as a single unit.
3. Apply algebraic squaring: Square the given expression \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2\) to create terms involving \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\), then isolate the target expression.
4. Calculate the final result: Substitute the known value and solve for \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find. We're given that \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2} = 4\), and we need to find \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\).

Think of this like a puzzle where we have one piece of information and need to use it to find another related piece. The key insight is that these expressions are connected - if we know something about \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2}\), we should be able to figure out something about \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\).

Process Skill: TRANSLATE - Converting the given condition into a form we can work with algebraically

2. Establish the connection pattern

Here's the crucial insight: notice that \(\mathrm{x}^4\) is the same as \((\mathrm{x}^2)^2\), and \(\frac{1}{\mathrm{x}^4}\) is the same as \((\frac{1}{\mathrm{x}^2})^2\). This means we can think of \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\) as \((\mathrm{x}^2)^2 + (\frac{1}{\mathrm{x}^2})^2\).

So if we let \(\mathrm{y} = \mathrm{x}^2\) and \(\mathrm{z} = \frac{1}{\mathrm{x}^2}\), then:

• We know that \(\mathrm{y} + \mathrm{z} = 4\) (from our given condition)
• We want to find \(\mathrm{y}^2 + \mathrm{z}^2\) (which is our target expression)

This transforms our problem from working with fourth powers to working with squares, which is much more manageable.

3. Apply algebraic squaring

Now we use a fundamental algebraic relationship. When we square a sum, we get:

\((\mathrm{y} + \mathrm{z})^2 = \mathrm{y}^2 + 2\mathrm{yz} + \mathrm{z}^2\)

Substituting back our expressions:

\((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = (\mathrm{x}^2)^2 + 2(\mathrm{x}^2)(\frac{1}{\mathrm{x}^2}) + (\frac{1}{\mathrm{x}^2})^2\)

\((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = \mathrm{x}^4 + 2 + \frac{1}{\mathrm{x}^4}\)

Notice that the middle term \(2(\mathrm{x}^2)(\frac{1}{\mathrm{x}^2}) = 2(1) = 2\), since \(\frac{\mathrm{x}^2}{\mathrm{x}^2} = 1\).

So we have: \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = \mathrm{x}^4 + \frac{1}{\mathrm{x}^4} + 2\)

Rearranging to isolate what we want:

\(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = (\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 - 2\)

Process Skill: MANIPULATE - Using algebraic identities to connect the given expression to our target

4. Calculate the final result

Now we substitute the known value:

\(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = (4)^2 - 2\)

\(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = 16 - 2\)

\(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = 14\)

Let's verify this makes sense: we started with \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2} = 4\), and found that \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = 14\). Since we're dealing with fourth powers instead of second powers, it's reasonable that our answer is larger than 4.

Final Answer

The value of \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\) is 14.

Looking at our answer choices: A. 2, B. 4, C. 6, D. 14, E. 16

Our answer matches choice D. 14.

Common Faltering Points

Errors while devising the approach

1. Attempting to solve for x first
Many students see \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2} = 4\) and immediately try to solve for the actual values of x. They might set up a quadratic equation or try to find specific values like \(\mathrm{x} = ±\sqrt{2}\). This approach, while mathematically possible, is unnecessarily complex and time-consuming. The elegant solution recognizes that we don't need to know x's value - we can work directly with the given expression.

2. Missing the algebraic relationship between the expressions
Students often fail to see that \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\) can be written as \((\mathrm{x}^2)^2 + (\frac{1}{\mathrm{x}^2})^2\), which directly relates to the given condition \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2} = 4\). Without this key insight, they may attempt various unproductive approaches or even conclude the problem cannot be solved with the given information.

3. Not recognizing the squaring identity pattern
Even if students see the connection between \(\mathrm{x}^2 + \frac{1}{\mathrm{x}^2}\) and \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\), they may not realize they should use the identity \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\). Instead, they might try to manipulate the expressions in other ways, leading to dead ends.

Errors while executing the approach

1. Algebraic manipulation errors in the squaring formula
When applying \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = (\mathrm{x}^2)^2 + 2(\mathrm{x}^2)(\frac{1}{\mathrm{x}^2}) + (\frac{1}{\mathrm{x}^2})^2\), students commonly make errors in the middle term. They might incorrectly calculate \(2(\mathrm{x}^2)(\frac{1}{\mathrm{x}^2})\) as something other than 2, or forget this term entirely, leading to \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = \mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\) instead of \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = \mathrm{x}^4 + \frac{1}{\mathrm{x}^4} + 2\).

2. Sign errors in rearrangement
When rearranging \((\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 = \mathrm{x}^4 + \frac{1}{\mathrm{x}^4} + 2\) to isolate \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4}\), students may incorrectly write \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = (\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 + 2\) instead of \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = (\mathrm{x}^2 + \frac{1}{\mathrm{x}^2})^2 - 2\). This sign error would lead to the incorrect answer of 18.

Errors while selecting the answer

1. Arithmetic calculation mistakes
Even with the correct setup \(\mathrm{x}^4 + \frac{1}{\mathrm{x}^4} = (4)^2 - 2\), students may make simple arithmetic errors. They might calculate \(4^2\) as 8 instead of 16, or make errors in the final subtraction \(16 - 2\), potentially arriving at incorrect values like 12 or 16. These computational slips can lead them to select wrong answer choices.