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If \(\mathrm{k} \neq 0\) and \(\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}} = \frac{\mathrm{x}}{\mathrm{k}}\), then \(\mathrm{x} =\)
Let's start by understanding what this equation is telling us in plain English. We have an equation that says: "k minus some fraction equals another fraction."
The equation is: \(\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}} = \frac{\mathrm{x}}{\mathrm{k}}\)
What we're looking for is the value of x in terms of k. Think of this like a word problem where we need to find what number x represents when we know the value of k.
The key insight here is that we have fractions on both sides, and since \(\mathrm{k} ≠ 0\), we can work with these fractions safely without worrying about division by zero.
Process Skill: TRANSLATE - Converting the algebraic equation into understandable language helps us see what we're solving for.
The most effective way to handle an equation with fractions is to get rid of the fractions entirely. Since both sides of our equation involve the denominator k, we can multiply everything by k.
Starting with: \(\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}} = \frac{\mathrm{x}}{\mathrm{k}}\)
When we multiply both sides by k, here's what happens:
• Left side: \(\mathrm{k} × [\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}}] = \mathrm{k} × \mathrm{k} - \mathrm{k} × \frac{3 - 2\mathrm{k}^2}{\mathrm{k}} = \mathrm{k}^2 - (3 - 2\mathrm{k}^2)\)
• Right side: \(\mathrm{k} × \frac{\mathrm{x}}{\mathrm{k}} = \mathrm{x}\)
So our equation becomes: \(\mathrm{k}^2 - (3 - 2\mathrm{k}^2) = \mathrm{x}\)
Notice how much cleaner this looks! We've eliminated all the fractions and now have a straightforward algebraic expression.
Now we need to simplify the left side of our equation. We have: \(\mathrm{k}^2 - (3 - 2\mathrm{k}^2) = \mathrm{x}\)
Let's think about what "minus a quantity in parentheses" means. When we subtract \((3 - 2\mathrm{k}^2)\), we need to distribute the negative sign to everything inside the parentheses.
\(\mathrm{k}^2 - (3 - 2\mathrm{k}^2)\) becomes:
\(\mathrm{k}^2 - 3 + 2\mathrm{k}^2\)
Now we can combine like terms. We have \(\mathrm{k}^2\) and \(2\mathrm{k}^2\) which are both \(\mathrm{k}^2\) terms:
\(\mathrm{k}^2 + 2\mathrm{k}^2 - 3 = 3\mathrm{k}^2 - 3\)
Therefore: \(\mathrm{x} = 3\mathrm{k}^2 - 3\)
Process Skill: SIMPLIFY - Careful distribution of the negative sign and combining like terms is crucial for getting the correct final answer.
Our solution shows that \(\mathrm{x} = 3\mathrm{k}^2 - 3\).
Looking at the answer choices:
Our answer \(\mathrm{x} = 3\mathrm{k}^2 - 3\) matches choice (C) exactly.
We can verify this makes sense by checking our algebra: we started with fractions, cleared them by multiplying by k, then carefully distributed the negative sign and combined like terms. Each step follows logically from the previous one.
Answer: (C) \(3\mathrm{k}^2 - 3\)
1. Misunderstanding the constraint \(\mathrm{k} \neq 0\): Students may overlook that \(\mathrm{k} \neq 0\) is given specifically to ensure we can safely multiply both sides by k and work with fractions involving k. Without recognizing this constraint, they might worry unnecessarily about undefined expressions or choose a more complicated approach.
2. Attempting to solve for k instead of x: The equation structure with multiple instances of k might confuse students into thinking they need to find the value of k first, when the goal is actually to express x in terms of k. This leads to unnecessary complexity and wrong direction.
3. Avoiding fraction elimination: Some students might try to work directly with the fractions instead of recognizing that multiplying both sides by k is the most efficient approach. This leads to much more complicated algebraic manipulation and higher chance of errors.
1. Incorrect distribution of negative sign: When simplifying \(\mathrm{k}^2 - (3 - 2\mathrm{k}^2)\), students commonly make the error of not properly distributing the negative sign, writing \(\mathrm{k}^2 - 3 - 2\mathrm{k}^2\) instead of \(\mathrm{k}^2 - 3 + 2\mathrm{k}^2\). This sign error leads to \(-\mathrm{k}^2 - 3\) instead of the correct \(3\mathrm{k}^2 - 3\).
2. Multiplication errors when clearing fractions: When multiplying both sides by k, students might incorrectly handle \(\mathrm{k} × [\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}}]\), either forgetting to distribute k to both terms or making errors in the multiplication \(\mathrm{k} × \frac{3 - 2\mathrm{k}^2}{\mathrm{k}}\), not recognizing it simplifies to \((3 - 2\mathrm{k}^2)\).
3. Incorrect combining of like terms: After getting \(\mathrm{k}^2 - 3 + 2\mathrm{k}^2\), students might incorrectly combine the \(\mathrm{k}^2\) terms, perhaps adding \(\mathrm{k}^2 + 2\mathrm{k}^2 = 2\mathrm{k}^2\) instead of \(3\mathrm{k}^2\), leading to the wrong final expression \(2\mathrm{k}^2 - 3\).
1. Choosing answer choice (B) due to sign confusion: If students made the sign error during distribution, they might arrive at \(\mathrm{k}^2 - 3\) and select choice (B), not realizing they dropped or incorrectly handled the \(2\mathrm{k}^2\) term during their algebraic manipulation.
2. Not recognizing equivalent expressions: Students might arrive at the correct algebraic result but write it in a different order like \(-3 + 3\mathrm{k}^2\) and not recognize this is the same as \(3\mathrm{k}^2 - 3\), potentially leading them to think their answer doesn't match any of the choices.
Step 1: Choose a convenient value for k
Let's choose \(\mathrm{k} = 2\) (a simple non-zero value that will make calculations manageable)
Step 2: Substitute \(\mathrm{k} = 2\) into the original equation
\(\mathrm{k} - \frac{3 - 2\mathrm{k}^2}{\mathrm{k}} = \frac{\mathrm{x}}{\mathrm{k}}\) becomes:
\(2 - \frac{3 - 2(2)^2}{2} = \frac{\mathrm{x}}{2}\)
\(2 - \frac{3 - 2(4)}{2} = \frac{\mathrm{x}}{2}\)
\(2 - \frac{3 - 8}{2} = \frac{\mathrm{x}}{2}\)
\(2 - \frac{-5}{2} = \frac{\mathrm{x}}{2}\)
\(2 - (-2.5) = \frac{\mathrm{x}}{2}\)
\(2 + 2.5 = \frac{\mathrm{x}}{2}\)
\(4.5 = \frac{\mathrm{x}}{2}\)
\(\mathrm{x} = 9\)
Step 3: Test each answer choice with \(\mathrm{k} = 2\)
(A) \(-3 - \mathrm{k}^2 = -3 - (2)^2 = -3 - 4 = -7\) ✗
(B) \(\mathrm{k}^2 - 3 = (2)^2 - 3 = 4 - 3 = 1\) ✗
(C) \(3\mathrm{k}^2 - 3 = 3(2)^2 - 3 = 3(4) - 3 = 12 - 3 = 9\) ✓
(D) \(\mathrm{k} - 3 - 2\mathrm{k}^2 = 2 - 3 - 2(4) = 2 - 3 - 8 = -9\) ✗
(E) \(\mathrm{k} - 3 + 2\mathrm{k}^2 = 2 - 3 + 2(4) = 2 - 3 + 8 = 7\) ✗
Step 4: Verify with another value
Let's try \(\mathrm{k} = 1\) to confirm:
\(1 - \frac{3 - 2(1)^2}{1} = \frac{\mathrm{x}}{1}\)
\(1 - \frac{3 - 2}{1} = \mathrm{x}\)
\(1 - 1 = \mathrm{x}\)
\(\mathrm{x} = 0\)
Testing choice (C): \(3\mathrm{k}^2 - 3 = 3(1)^2 - 3 = 3 - 3 = 0\) ✓
Conclusion: The answer is (C) \(3\mathrm{k}^2 - 3\)