Simplify The Expression:${ (-2x^2 + 8x - 8) + (-7x + 9) }$

Understanding the Problem

When dealing with algebraic expressions, it's essential to simplify them to make calculations easier and more manageable. In this article, we'll focus on simplifying the given expression by combining like terms. Like terms are terms that have the same variable raised to the same power. By combining these terms, we can rewrite the expression in a simpler form.

The Given Expression

The given expression is:

$$(-2x^2 + 8x - 8) + (-7x + 9)$$

Combining Like Terms

To simplify the expression, we need to combine the like terms. The first step is to identify the like terms in the expression. In this case, the like terms are the terms with the variable $x$ raised to the same power.

Combining the Constant Terms

The constant terms in the expression are $-8$ and $9$. We can combine these terms by adding them together:

$$-8 + 9 = 1$$

So, the constant term in the simplified expression is $1$.

Combining the Terms with the Variable $x$

The terms with the variable $x$ are $8x$ and $-7x$. We can combine these terms by adding their coefficients:

$$8x - 7x = x$$

So, the term with the variable $x$ in the simplified expression is $x$.

Combining the Terms with the Variable $x^2$

The term with the variable $x^2$ is $-2x^2$. Since there are no other terms with the variable $x^2$, this term remains the same in the simplified expression.

Simplifying the Expression

Now that we have combined the like terms, we can rewrite the expression in a simpler form:

$$-2x^2 + x + 1$$

This is the simplified expression.

Conclusion

In this article, we simplified the given expression by combining like terms. We identified the like terms, combined the constant terms, and combined the terms with the variable $x$. By following these steps, we were able to rewrite the expression in a simpler form. This process is essential in algebra, as it makes calculations easier and more manageable.

Tips for Simplifying Expressions

When simplifying expressions, it's essential to follow these tips:

• Identify the like terms in the expression.
• Combine the constant terms by adding them together.
• Combine the terms with the variable by adding their coefficients.
• Leave the terms with the variable raised to different powers unchanged.

By following these tips, you can simplify expressions and make calculations easier.

Examples of Simplifying Expressions

Here are some examples of simplifying expressions:

Example 1

Simplify the expression: $(-3x^2 + 4x - 2) + (2x^2 - 3x + 1)$

To simplify this expression, we need to combine the like terms. The like terms are the terms with the variable $x^2$ and the terms with the variable $x$.

The constant terms are $-2$ and $1$. We can combine these terms by adding them together:

$$-2 + 1 = -1$$

So, the constant term in the simplified expression is $-1$.

The terms with the variable $x^2$ are $-3x^2$ and $2x^2$. We can combine these terms by adding their coefficients:

$$-3x^2 + 2x^2 = -x^2$$

So, the term with the variable $x^2$ in the simplified expression is $-x^2$.

The terms with the variable $x$ are $4x$ and $-3x$. We can combine these terms by adding their coefficients:

$$4x - 3x = x$$

So, the term with the variable $x$ in the simplified expression is $x$.

The simplified expression is:

$$-x^2 + x - 1$$

Example 2

Simplify the expression: $(2x^2 + 3x - 1) + (-4x^2 + 2x + 3)$

To simplify this expression, we need to combine the like terms. The like terms are the terms with the variable $x^2$ and the terms with the variable $x$.

The constant terms are $-1$ and $3$. We can combine these terms by adding them together:

$$-1 + 3 = 2$$

So, the constant term in the simplified expression is $2$.

The terms with the variable $x^2$ are $2x^2$ and $-4x^2$. We can combine these terms by adding their coefficients:

$$2x^2 - 4x^2 = -2x^2$$

So, the term with the variable $x^2$ in the simplified expression is $-2x^2$.

The terms with the variable $x$ are $3x$ and $2x$. We can combine these terms by adding their coefficients:

$$3x + 2x = 5x$$

So, the term with the variable $x$ in the simplified expression is $5x$.

The simplified expression is:

$$-2x^2 + 5x + 2$$

Conclusion

In this article, we simplified two expressions by combining like terms. We identified the like terms, combined the constant terms, and combined the terms with the variable $x$. By following these steps, we were able to rewrite the expressions in simpler forms. This process is essential in algebra, as it makes calculations easier and more manageable.

Final Tips

When simplifying expressions, it's essential to follow these final tips:

• Always identify the like terms in the expression.
• Combine the constant terms by adding them together.
• Combine the terms with the variable by adding their coefficients.
• Leave the terms with the variable raised to different powers unchanged.

By following these tips, you can simplify expressions and make calculations easier.

Frequently Asked Questions

In this article, we'll answer some frequently asked questions about simplifying expressions by combining like terms.

Q: What are like terms in algebra?

A: Like terms in algebra are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I identify like terms in an expression?

A: To identify like terms in an expression, you need to look for terms that have the same variable raised to the same power. For example, in the expression $2x^2 + 3x + 4$, the like terms are $2x^2$ and $3x$ because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms in an expression?

A: To combine like terms in an expression, you need to add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 3x + 4$, the like terms are $2x^2$ and $3x$. To combine these terms, you need to add their coefficients:

$$2x^2 + 3x = (2+3)x^2 = 5x^2$$

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms is a step in simplifying an expression. When you combine like terms, you are reducing the number of terms in the expression by adding or subtracting their coefficients. Simplifying an expression, on the other hand, involves reducing the expression to its simplest form by combining like terms, removing any unnecessary parentheses, and rearranging the terms in a more convenient order.

Q: Can I simplify an expression by combining like terms if there are no like terms?

A: No, you cannot simplify an expression by combining like terms if there are no like terms. In this case, the expression is already in its simplest form, and there is no need to combine any terms.

Q: How do I know if an expression is already in its simplest form?

A: To determine if an expression is already in its simplest form, you need to check if there are any like terms that can be combined. If there are no like terms, then the expression is already in its simplest form.

Q: Can I simplify an expression by combining like terms if the expression has variables with different exponents?

A: No, you cannot simplify an expression by combining like terms if the expression has variables with different exponents. In this case, the terms with different exponents are not like terms, and you cannot combine them.

Q: How do I simplify an expression with variables with different exponents?

A: To simplify an expression with variables with different exponents, you need to leave the terms with different exponents unchanged. For example, in the expression $2x^2 + 3x + 4$, the term $2x^2$ has a different exponent than the term $3x$. In this case, you cannot combine these terms, and the expression is already in its simplest form.

Q: Can I simplify an expression by combining like terms if the expression has coefficients with different signs?

A: No, you cannot simplify an expression by combining like terms if the expression has coefficients with different signs. In this case, the terms with different signs are not like terms, and you cannot combine them.

Q: How do I simplify an expression with coefficients with different signs?

A: To simplify an expression with coefficients with different signs, you need to leave the terms with different signs unchanged. For example, in the expression $2x^2 - 3x + 4$, the term $2x^2$ has a positive coefficient, while the term $-3x$ has a negative coefficient. In this case, you cannot combine these terms, and the expression is already in its simplest form.

Q: Can I simplify an expression by combining like terms if the expression has parentheses?

A: Yes, you can simplify an expression by combining like terms if the expression has parentheses. In this case, you need to remove the parentheses and combine the like terms inside the parentheses.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, you need to remove the parentheses and combine the like terms inside the parentheses. For example, in the expression $(2x^2 + 3x + 4) + (5x^2 - 2x + 1)$, you need to remove the parentheses and combine the like terms:

$$(2x^2 + 3x + 4) + (5x^2 - 2x + 1) = 2x^2 + 3x + 4 + 5x^2 - 2x + 1$$

$$= 7x^2 + x + 5$$

Q: Can I simplify an expression by combining like terms if the expression has fractions?

A: Yes, you can simplify an expression by combining like terms if the expression has fractions. In this case, you need to combine the fractions and then combine the like terms.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to combine the fractions and then combine the like terms. For example, in the expression $\frac{2x^2}{3} + \frac{3x}{4} + \frac{1}{2}$, you need to combine the fractions and then combine the like terms:

$$\frac{2x^2}{3} + \frac{3x}{4} + \frac{1}{2} = \frac{8x^2}{12} + \frac{9x}{12} + \frac{6}{12}$$

$$= \frac{8x^2 + 9x + 6}{12}$$

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions by combining like terms. We discussed the importance of identifying like terms, combining like terms, and simplifying expressions. We also provided examples of simplifying expressions with variables with different exponents, coefficients with different signs, parentheses, and fractions. By following these steps, you can simplify expressions and make calculations easier.