Applying the Rules of Subtraction

Oct 13, 2025 by pinkandtrash.com 5 views

Understanding the Problem

When dealing with algebraic expressions, we often need to perform operations such as addition and subtraction. In this problem, we are asked to subtract one expression from another. To do this, we need to apply the rules of subtraction, which involve changing the signs of the terms in the second expression.

Applying the Rules of Subtraction

To subtract $\left(-2 f^2-f+3\right)$ from $\left(6 f^2-9 f+10\right)$ , we need to change the signs of the terms in the second expression. This means that we will change the signs of the coefficients of the terms in the second expression.

Step 1: Change the Signs of the Terms in the Second Expression

The second expression is $\left(-2 f^2-f+3\right)$ . To change the signs of the terms, we will multiply each term by $-1$ . This gives us:

\left(-1\right)\left(-2 f^2\right) + \left(-1\right)\left(-f\right) + \left(-1\right)\left(3\right)

Step 2: Simplify the Expression

Now that we have changed the signs of the terms, we can simplify the expression by combining like terms. This gives us:

2 f^2 + f - 3

Step 3: Subtract the Second Expression from the First Expression

Now that we have simplified the second expression, we can subtract it from the first expression. This gives us:

\left(6 f^2-9 f+10\right) - \left(2 f^2 + f - 3\right)

Step 4: Combine Like Terms

To combine like terms, we need to distribute the negative sign to each term in the second expression. This gives us:

6 f^2 - 9 f + 10 - 2 f^2 - f + 3

Step 5: Simplify the Expression

Now that we have distributed the negative sign, we can combine like terms. This gives us:

\left(6 f^2 - 2 f^2\right) + \left(-9 f - f\right) + \left(10 + 3\right)

Step 6: Simplify Further

Simplifying further, we get:

4 f^2 - 10 f + 13

Conclusion

Therefore, the correct answer is:

$4 f^2-10 f+13$

This is the result of subtracting $\left(-2 f^2-f+3\right)$ from $\left(6 f^2-9 f+10\right)$ .

Comparison with the Options

Let's compare our result with the options given:

Option A: $4 f^2-10 f+7$
Option B: $8 f^2-10 f+7$
Option C: $8 f^2-8 f+7$
Option D: $4 f^2-10 f+13$

Our result matches option D: $4 f^2-10 f+13$ .

Final Answer

The final answer is $\boxed{D}$ .

Discussion

This problem requires the application of the rules of subtraction to simplify an algebraic expression. The key concept is to change the signs of the terms in the second expression and then combine like terms. This problem is a good example of how to apply the rules of subtraction in algebra.

Tips and Tricks

When subtracting one expression from another, make sure to change the signs of the terms in the second expression.
Combine like terms to simplify the expression.
Make sure to distribute the negative sign to each term in the second expression.

Common Mistakes

Failing to change the signs of the terms in the second expression.
Not combining like terms.
Not distributing the negative sign to each term in the second expression.

Real-World Applications

This problem has real-world applications in algebra and mathematics. It can be used to model real-world situations, such as calculating the difference between two quantities.

Conclusion

In conclusion, this problem requires the application of the rules of subtraction to simplify an algebraic expression. The key concept is to change the signs of the terms in the second expression and then combine like terms. This problem is a good example of how to apply the rules of subtraction in algebra.

Frequently Asked Questions

Q: What is the correct answer to the problem?

A: The correct answer is $4 f^2-10 f+13$ .

Q: How do I change the signs of the terms in the second expression?

A: To change the signs of the terms in the second expression, you need to multiply each term by $-1$ . This will give you the opposite of the original expression.

Q: What is the difference between subtracting and adding expressions?

A: When subtracting expressions, you need to change the signs of the terms in the second expression. When adding expressions, you do not need to change the signs of the terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{D}$ .

Q: Can you explain the steps to solve the problem?

A: Here are the steps to solve the problem:

Change the signs of the terms in the second expression.
Simplify the second expression.
Subtract the second expression from the first expression.
Combine like terms.
Simplify the expression.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Failing to change the signs of the terms in the second expression.
Not combining like terms.
Not distributing the negative sign to each term in the second expression.

Q: How can I apply this problem to real-world situations?

A: This problem can be used to model real-world situations, such as calculating the difference between two quantities.

Q: What are some tips and tricks for solving this problem?

A: Some tips and tricks for solving this problem include:

Make sure to change the signs of the terms in the second expression.
Combine like terms to simplify the expression.
Make sure to distribute the negative sign to each term in the second expression.

Additional Resources

For more practice problems, try the following resources:

Khan Academy: Algebra
Mathway: Algebra
IXL: Algebra

For more information on algebra, try the following resources:

Khan Academy: Algebra
Mathway: Algebra
IXL: Algebra