Simplify The Expression:$\[ \left(5x^2 - 3x - 6\right) + \left(x^2 - 4x + 7\right) \\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common techniques used to simplify expressions is combining like terms. In this article, we will focus on simplifying the given expression by combining like terms.

The Expression to Simplify

The given expression is:

$$\left(5x^2 - 3x - 6\right) + \left(x^2 - 4x + 7\right)$$

Understanding Like Terms

Like terms are terms that have the same variable raised to the same power. In other words, they have the same exponent. For example, $x^2$ and $x^2$ are like terms, while $x^2$ and $x$ are not.

Step 1: Identify Like Terms

To simplify the given expression, we need to identify the like terms. In this case, we have two sets of like terms:

• $5x^2$ and $x^2$
• $-3x$ and $-4x$
• $-6$ and $7$

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them. To do this, we add or subtract the coefficients of the like terms.

• For the terms $5x^2$ and $x^2$, we add their coefficients: $5 + 1 = 6$. So, the combined term is $6x^2$.
• For the terms $-3x$ and $-4x$, we add their coefficients: $-3 + (-4) = -7$. So, the combined term is $-7x$.
• For the terms $-6$ and $7$, we add them: $-6 + 7 = 1$. So, the combined term is $1$.

Simplifying the Expression

Now that we have combined the like terms, we can simplify the expression by adding the combined terms.

$$\left(5x^2 - 3x - 6\right) + \left(x^2 - 4x + 7\right) = 6x^2 - 7x + 1$$

Conclusion

In this article, we simplified the given expression by combining like terms. We identified the like terms, combined them, and then simplified the expression. This technique is essential in algebra and is used to solve equations and inequalities.

Tips and Tricks

• When combining like terms, make sure to add or subtract the coefficients correctly.
• Use parentheses to group like terms together.
• Simplify the expression by combining like terms.

Common Mistakes to Avoid

• Not identifying like terms correctly.
• Not combining like terms correctly.
• Not simplifying the expression after combining like terms.

Real-World Applications

Combining like terms is a crucial skill in algebra and has many real-world applications. For example, in physics, combining like terms is used to simplify equations of motion. In engineering, combining like terms is used to simplify equations of electrical circuits.

Practice Problems

1. Simplify the expression: $\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)$
2. Simplify the expression: $\left(4x^2 - 2x + 3\right) + \left(2x^2 + 4x - 1\right)$

Answer Key

1. $3x^2 + x + 3$
2. $6x^2 + 2x + 2$

Conclusion

Introduction

In our previous article, we discussed how to simplify expressions by combining like terms. In this article, we will provide a Q&A guide to help you understand the concept better.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. In other words, they have the same exponent. For example, $x^2$ and $x^2$ are like terms, while $x^2$ and $x$ are not.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable raised to the same power. For example, in the expression $2x^2 + 3x - 1$, the like terms are $2x^2$ and $3x$.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 3x - 1$, the like terms are $2x^2$ and $3x$. To combine them, add their coefficients: $2 + 3 = 5$. So, the combined term is $5x^2$.

Q: What if I have multiple sets of like terms?

A: If you have multiple sets of like terms, combine them separately and then add or subtract the results. For example, in the expression $2x^2 + 3x - 1 + x^2 - 2x + 4$, the like terms are $2x^2$ and $x^2$, and $3x$ and $-2x$. To combine them, add their coefficients: $2 + 1 = 3$ and $3 + (-2) = 1$. So, the combined terms are $3x^2$ and $x$.

Q: Can I simplify an expression with variables and constants?

A: Yes, you can simplify an expression with variables and constants by combining like terms. For example, in the expression $2x^2 + 3x - 1 + x^2 - 2x + 4$, the like terms are $2x^2$ and $x^2$, and $3x$ and $-2x$. To combine them, add their coefficients: $2 + 1 = 3$ and $3 + (-2) = 1$. So, the combined terms are $3x^2$ and $x$.

Q: What if I have a negative coefficient?

A: If you have a negative coefficient, simply add or subtract the coefficient as usual. For example, in the expression $-2x^2 + 3x - 1$, the like terms are $-2x^2$ and $3x$. To combine them, add their coefficients: $-2 + 3 = 1$. So, the combined term is $x^2$.

Q: Can I simplify an expression with fractions?

A: Yes, you can simplify an expression with fractions by combining like terms. For example, in the expression $\frac{1}{2}x^2 + \frac{3}{4}x - \frac{1}{2}$, the like terms are $\frac{1}{2}x^2$ and $-\frac{1}{2}$. To combine them, add their coefficients: $\frac{1}{2} + (-\frac{1}{2}) = 0$. So, the combined term is $0$.

Q: What if I have a variable with a coefficient of 1?

A: If you have a variable with a coefficient of 1, you can simply drop the coefficient and keep the variable. For example, in the expression $1x^2 + 3x - 1$, the like terms are $x^2$ and $3x$. To combine them, add their coefficients: $1 + 3 = 4$. So, the combined term is $4x^2$.

Conclusion

In conclusion, simplifying expressions by combining like terms is a crucial skill in algebra. By identifying like terms, combining them, and simplifying the expression, we can solve equations and inequalities. Remember to use parentheses to group like terms together and simplify the expression after combining like terms. With practice, you will become proficient in simplifying expressions and solving equations and inequalities.

Practice Problems

1. Simplify the expression: $\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)$
2. Simplify the expression: $\left(4x^2 - 2x + 3\right) + \left(2x^2 + 4x - 1\right)$
3. Simplify the expression: $\frac{1}{2}x^2 + \frac{3}{4}x - \frac{1}{2}$

Answer Key

1. $3x^2 + x + 3$
2. $6x^2 + 2x + 2$
3. $0$