Simplify: (-g-1) + (-3g^2 + 6g) - Math Problem Solved!

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Alright, let's break down how to simplify the expression $(-g-1) + (-3g^2 + 6g)$. This is a classic algebra problem where we need to combine like terms. Don't worry; it's easier than it looks! We'll go through it step by step, so you'll be a pro in no time. First, I'll provide a detailed walkthrough; then, I will provide some extra information. Also, let's think about why simplifying expressions like this is super useful in real life. When you get the hang of it, it's like having a superpower for solving problems. Remember, math is all about practice, so the more you do, the better you'll get. Let's get started, guys!

Step-by-Step Solution

First, rewrite the expression:

$(-g-1) + (-3g^2 + 6g)$

Remove parentheses:

$-g - 1 - 3g^2 + 6g$

Now, group the like terms:

$-3g^2 + (-g + 6g) - 1$

Combine the 'g' terms:

$-3g^2 + 5g - 1$

So, the simplified expression is:

$-3g^2 + 5g - 1$

Detailed Explanation

1. Original Expression

The expression we start with is $(-g-1) + (-3g^2 + 6g)$. The goal is to simplify this by combining terms that are alike. Think of it like organizing your closet: you want to put all the shirts together, all the pants together, and so on. In math, we group terms with the same variable and exponent together.

2. Removing Parentheses

When we remove the parentheses, we need to be careful with the signs. Since we are adding the two expressions, the signs inside the parentheses stay the same. So, $(-g-1)$ becomes $-g - 1$, and $(-3g^2 + 6g)$ becomes $-3g^2 + 6g$. This gives us: $-g - 1 - 3g^2 + 6g$.

3. Grouping Like Terms

Now, let's group the like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have terms with $g^2$, terms with $g$, and constant terms (numbers without any variables). So we rearrange the expression to group these together: $-3g^2 + (-g + 6g) - 1$. Notice how we keep the signs the same as we rearrange.

4. Combining Like Terms

Next, we combine the like terms. We have $-g + 6g$, which simplifies to $5g$. So our expression becomes $-3g^2 + 5g - 1$. There are no other like terms to combine, so we are done!

5. Final Simplified Expression

The simplified expression is $-3g^2 + 5g - 1$. This is a quadratic expression, and it's now in its simplest form. We can't simplify it any further unless we have additional information or instructions, such as solving for $g$.

Why is This Important?

You might be wondering, why do we even bother simplifying expressions? Well, simplifying expressions is a fundamental skill in algebra and is used in many real-world applications. For example, when you're calculating the area or volume of a shape, you often end up with an expression that needs to be simplified. Or, if you're writing a computer program, you might need to simplify an expression to make your code more efficient. Let's look at a few practical examples:

1. Calculating Areas and Volumes

Imagine you're designing a rectangular garden. The length of the garden is $(2x + 3)$ feet, and the width is $(x - 1)$ feet. To find the area of the garden, you multiply the length by the width: $(2x + 3)(x - 1)$. Expanding this gives you $2x^2 + x - 3$. This is a simplified expression that tells you how the area of the garden depends on the value of $x$.

2. Physics Problems

In physics, you often encounter expressions that need to be simplified. For example, if you're calculating the kinetic energy of an object, you might have an expression like $\frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity. If you have multiple terms involving $m$ and $v$, you'll need to simplify the expression to find the total kinetic energy.

3. Computer Programming

In computer programming, simplifying expressions can make your code run faster and more efficiently. For example, if you have a complex calculation that is repeated many times in your program, simplifying the expression can reduce the amount of time it takes to run the program.

Additional Tips and Tricks

1. Practice Regularly

The best way to get better at simplifying expressions is to practice regularly. Start with simple problems and gradually work your way up to more complex ones. The more you practice, the more comfortable you'll become with the process.

2. Use Online Resources

There are many online resources available to help you practice simplifying expressions. Websites like Khan Academy and Mathway offer practice problems, video tutorials, and step-by-step solutions. These resources can be a great way to supplement your learning.

3. Break Down Complex Problems

When you encounter a complex problem, break it down into smaller, more manageable steps. This will make the problem less intimidating and easier to solve. Write down each step as you go, so you can keep track of your progress.

4. Check Your Work

Always check your work to make sure you haven't made any mistakes. One way to do this is to plug in a value for the variable and see if the original expression and the simplified expression give you the same answer. If they don't, you've made a mistake somewhere.

5. Stay Organized

Keep your work organized by writing neatly and lining up like terms. This will make it easier to spot mistakes and keep track of your progress. Use different colors to highlight like terms, if that helps you stay organized.

Common Mistakes to Avoid

1. Forgetting to Distribute

One common mistake is forgetting to distribute a negative sign when removing parentheses. For example, if you have $-(x + 2)$, you need to distribute the negative sign to both terms inside the parentheses, so it becomes $-x - 2$. Make sure you pay close attention to the signs when removing parentheses.

2. Combining Unlike Terms

Another common mistake is combining unlike terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, you can't combine $x^2$ and $x$, because they are not like terms.

3. Making Sign Errors

Sign errors are very common, especially when dealing with negative numbers. Be careful when adding, subtracting, multiplying, and dividing negative numbers. Double-check your work to make sure you haven't made any sign errors.

4. Not Simplifying Completely

Make sure you simplify the expression completely. This means combining all like terms and reducing fractions to their simplest form. If you leave any like terms uncombined, you haven't fully simplified the expression.

5. Rushing Through the Problem

It's important to take your time and work carefully. Rushing through the problem can lead to mistakes. Read the problem carefully, identify the like terms, and combine them one step at a time.

Practice Problems

To help you practice, here are a few more problems you can try:

1. $(2x + 3) + (4x - 1)$
2. $(5y - 2) - (2y + 1)$
3. $(3a^2 + 2a - 1) + (a^2 - a + 3)$
4. $(4b^3 - b^2 + 2b) - (2b^3 + 3b^2 - b)$
5. $(-2c + 5) + (-4c - 3)$

Try solving these problems on your own, and then check your answers with a friend or online resource. Remember, the more you practice, the better you'll get!

Conclusion

So, to wrap things up, simplifying the expression $(-g-1) + (-3g^2 + 6g)$ involves removing parentheses, grouping like terms, and combining those like terms. The final simplified expression is $-3g^2 + 5g - 1$. This skill is super useful in many areas of math and science, so it's worth the effort to master it. Keep practicing, and you'll become a pro in no time! And remember, math can be fun when you approach it with a positive attitude and a willingness to learn. Keep up the great work, guys! We have explained the problem by using real-world examples, additional tips, and also common mistakes to avoid. I hope you found this helpful and that you feel more confident in your ability to simplify expressions. Happy learning!