Sharina's Simplification Error: Find And Fix!

by Alex Johnson 46 views

Let's dive into an interesting mathematical problem where we'll analyze Sharina's attempt to simplify the expression $3(2x - 6 - x + 1)^2 - 2 + 4x$. Sharina took two steps: first, simplifying within the parentheses, and second, expanding the exponent. Our mission is to meticulously examine her work, pinpoint any errors, and provide the correct simplification. This exercise is a fantastic way to sharpen our algebraic skills and understand the importance of following the order of operations.

Sharina's Attempted Simplification

To begin, let's take a look at Sharina's steps:

Original Expression:

3(2x−6−x+1)2−2+4x3(2x - 6 - x + 1)^2 - 2 + 4x

Step 1: Simplification within the parentheses.

This is where Sharina likely combined like terms inside the parentheses. We need to carefully retrace her steps to ensure accuracy. The original expression inside the parentheses is $2x - 6 - x + 1$. Combining the 'x' terms ($2x$ and $-x$) gives us $x$. Combining the constant terms ($-6$ and $+1$) gives us $-5$. So, the simplified expression inside the parentheses should be $x - 5$. Let's assume Sharina got this part right for now, but we'll need to verify it in the context of her complete solution.

Step 2: Expanding the exponent.

This is where the expression $(x - 5)^2$ needs to be expanded. Remember, squaring a binomial means multiplying it by itself: $(x - 5)(x - 5)$. Using the distributive property (or the FOIL method), we get: $x^2 - 5x - 5x + 25$, which simplifies to $x^2 - 10x + 25$. We need to compare this expansion to Sharina's result to identify any potential errors. Did she correctly apply the distributive property? Did she make any mistakes in combining like terms after the multiplication? These are crucial questions we need to answer.

Identifying Potential Errors

To effectively identify errors, we need to consider common pitfalls in algebraic simplification:

  • Incorrectly combining like terms: Did Sharina add or subtract terms with different variables or exponents? For example, did she try to combine an $x^2$ term with an $x$ term?
  • Mistakes in the distributive property: Did she correctly multiply each term inside the first set of parentheses by each term inside the second set of parentheses when expanding $(x - 5)^2$? A common error is to only square the first and last terms (e.g., writing $x^2 + 25$ instead of the correct expansion).
  • Sign errors: Did she make any mistakes with negative signs? This is a very common source of errors, especially when dealing with subtraction and the distributive property.
  • Order of operations: Did she follow the correct order of operations (PEMDAS/BODMAS)? This is crucial in ensuring the correct simplification. Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

By carefully considering these potential errors, we can approach the problem systematically and identify where Sharina went wrong.

Correcting Sharina's Work: A Step-by-Step Approach

Let's embark on the journey of simplifying the expression $3(2x - 6 - x + 1)^2 - 2 + 4x$ correctly, highlighting each step and the reasoning behind it. This will not only help us identify Sharina's mistake but also reinforce the fundamental principles of algebraic simplification.

Step 1: Simplify within the parentheses.

Our primary focus here is to combine the like terms inside the parentheses. Like terms are those that have the same variable raised to the same power. In the expression $2x - 6 - x + 1$, we have two types of terms: terms with the variable 'x' and constant terms.

  • Combining 'x' terms: We have $2x$ and $-x$. Combining these gives us $2x - x = x$.
  • Combining constant terms: We have $-6$ and $+1$. Combining these gives us $-6 + 1 = -5$.

Therefore, the expression inside the parentheses simplifies to $x - 5$. So far so good!

Step 2: Expand the exponent.

Now, we need to deal with the exponent. We have $(x - 5)^2$, which means $(x - 5)$ multiplied by itself: $(x - 5)(x - 5)$. To expand this, we use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last).

  • First: Multiply the first terms in each set of parentheses: $x * x = x^2$.
  • Outer: Multiply the outer terms: $x * -5 = -5x$.
  • Inner: Multiply the inner terms: $-5 * x = -5x$.
  • Last: Multiply the last terms: $-5 * -5 = 25$.

Now, we add all these results together: $x^2 - 5x - 5x + 25$. Notice the importance of the negative signs! The next step is to combine the like terms, which are the two $-5x$ terms. This gives us: $x^2 - 10x + 25$. This is the correct expansion of $(x - 5)^2$.

Step 3: Multiply by the coefficient.

Now we substitute the expanded form back into the original expression: $3(x^2 - 10x + 25) - 2 + 4x$. We need to multiply the entire expression inside the parentheses by 3. This again involves the distributive property.

  • 3 * x^2 = 3x^2$.

  • 3 * -10x = -30x$.

  • 3 * 25 = 75$.

So, $3(x^2 - 10x + 25)$ becomes $3x^2 - 30x + 75$.

Step 4: Combine all terms.

Now our expression looks like this: $3x^2 - 30x + 75 - 2 + 4x$. We need to combine all the like terms. We have terms with $x^2$, terms with $x$, and constant terms.

  • $x^2$ term: We only have one term with $x^2$, which is $3x^2$. So, it remains as it is.
  • $x$ terms: We have $-30x$ and $+4x$. Combining these gives us $-30x + 4x = -26x$.
  • Constant terms: We have $+75$ and $-2$. Combining these gives us $75 - 2 = 73$.

Final Simplified Expression:

Putting it all together, the simplified expression is: $3x^2 - 26x + 73$.

Where Did Sharina Go Wrong?

To pinpoint Sharina's error, we need to compare her steps (which, unfortunately, aren't explicitly provided) with our correct solution. It's highly probable that the error occurred during the expansion of the exponent or in the subsequent multiplication and combination of terms. Common mistakes include:

  • Incorrectly expanding $(x - 5)^2$: Perhaps Sharina missed the middle term (-10x) and wrote something like $x^2 + 25$.
  • Distributive property errors: She might have made a mistake while multiplying the expanded expression by 3.
  • Sign errors: A simple sign error can throw off the entire calculation.
  • Combining unlike terms: It's possible she mistakenly combined terms that couldn't be combined.

Without knowing her exact steps, we can only speculate. However, by understanding the correct process, we can recognize and avoid these common pitfalls in our own algebraic manipulations.

Key Takeaways and Best Practices

This exercise highlights several crucial aspects of algebraic simplification:

  • Order of Operations: Always adhere to the order of operations (PEMDAS/BODMAS). Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
  • Distributive Property: Master the distributive property. It's fundamental to expanding expressions correctly.
  • Combining Like Terms: Only combine terms that have the same variable raised to the same power.
  • Sign Awareness: Pay meticulous attention to signs, especially negative signs. They are a frequent source of errors.
  • Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This reduces the chance of errors and makes it easier to track your work.
  • Verification: If possible, verify your answer by plugging in a value for 'x' into both the original expression and the simplified expression. If the results don't match, there's an error.

By following these best practices, we can confidently tackle algebraic simplification problems and minimize the risk of making mistakes like Sharina might have.

Conclusion

Simplifying algebraic expressions requires careful attention to detail and a solid understanding of fundamental principles. By meticulously working through each step, we not only arrive at the correct answer but also deepen our understanding of the underlying concepts. Analyzing potential errors, like the ones Sharina might have made, helps us develop a more robust approach to problem-solving in mathematics. Remember, practice makes perfect, and the more we engage with these types of problems, the more proficient we become!

For further exploration and practice with algebraic expressions, check out resources like Khan Academy's Algebra 1 section, which offers comprehensive lessons and exercises.