What Is The Difference Of The Following Expression?${ 9n^2 - 2n }$2) Which Expression Represents The Sum Of ${ 3x^2 + 9x - 8 }$ And ${ -5(x^2 + 8 - 3x) }$?A. { -2x^2 + 6x$} B . \[ B. \[ B . \[ -2x^2 - 6x +

What is the Difference of Two Algebraic Expressions?

In algebra, we often encounter expressions that involve variables, constants, and mathematical operations. Understanding the difference between two algebraic expressions is crucial in solving equations, graphing functions, and simplifying complex expressions. In this article, we will explore the difference between two given expressions and then move on to finding the sum of two other algebraic expressions.

Expression 1: $9n^2 - 2n$

The first expression is $9n^2 - 2n$. This expression involves a quadratic term, $9n^2$, and a linear term, $-2n$. To find the difference of this expression, we need to understand that the difference between two expressions is obtained by subtracting one expression from the other.

Expression 2: $3x^2 + 9x - 8$

The second expression is $3x^2 + 9x - 8$. This expression involves a quadratic term, $3x^2$, a linear term, $9x$, and a constant term, $-8$.

Which Expression Represents the Sum of $3x^2 + 9x - 8$ and $-5(x^2 + 8 - 3x)$?

To find the sum of two expressions, we need to add the corresponding terms of the two expressions. Let's start by simplifying the second expression, $-5(x^2 + 8 - 3x)$.

Simplifying the Second Expression

We can simplify the second expression by distributing the negative sign to each term inside the parentheses:

$$-5(x^2 + 8 - 3x) = -5x^2 - 40 + 15x$$

Now, we can rewrite the second expression as:

$$-5x^2 - 40 + 15x$$

Finding the Sum of the Two Expressions

To find the sum of the two expressions, we need to add the corresponding terms:

$$(3x^2 + 9x - 8) + (-5x^2 - 40 + 15x)$$

We can combine like terms by adding the coefficients of the same variables:

$$3x^2 - 5x^2 + 9x + 15x - 8 - 40$$

Simplifying the expression, we get:

$$-2x^2 + 24x - 48$$

However, we need to compare this result with the given options. Let's rewrite the expression in a more simplified form:

$$-2x^2 + 24x - 48 = -2x^2 + 24x - 48$$

Now, let's compare this result with the given options:

A. $-2x^2 + 6x$

This option is incorrect because the coefficient of the linear term is different.

B. $-2x^2 - 6x + 48$

This option is also incorrect because the coefficient of the linear term is different and the constant term is not the same.

C. $-2x^2 + 24x - 48$

This option is correct because it matches our result.

Conclusion

In this article, we explored the difference between two algebraic expressions and then moved on to finding the sum of two other algebraic expressions. We simplified the second expression and then added the corresponding terms to find the sum. Our result matched option C, which is the correct answer.

Key Takeaways

• The difference between two algebraic expressions is obtained by subtracting one expression from the other.
• To find the sum of two expressions, we need to add the corresponding terms.
• We can simplify expressions by combining like terms and distributing negative signs.

Practice Problems

1. Find the difference between the expressions $2x^2 + 5x - 3$ and $x^2 - 2x + 1$.
2. Find the sum of the expressions $3x^2 + 2x - 4$ and $-2x^2 - 3x + 5$.

Solutions

1. The difference between the expressions $2x^2 + 5x - 3$ and $x^2 - 2x + 1$ is:

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

   Simplifying the expression, we get:

   $$x^2 + 7x - 4$$

2. The sum of the expressions $3x^2 + 2x - 4$ and $-2x^2 - 3x + 5$ is:

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

   Simplifying the expression, we get:

   x^2 - x + 1$<br/>

Q: What is the difference between two algebraic expressions?

A: The difference between two algebraic expressions is obtained by subtracting one expression from the other. For example, if we have two expressions $a$ and $b$, the difference between them is $a - b$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, we need to combine like terms and distribute negative signs. Like terms are terms that have the same variable raised to the same power. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: What is the sum of two algebraic expressions?

A: The sum of two algebraic expressions is obtained by adding the corresponding terms. For example, if we have two expressions $a$ and $b$, the sum of them is $a + b$.

Q: How do I distribute a negative sign to an algebraic expression?

A: To distribute a negative sign to an algebraic expression, we need to multiply each term inside the parentheses by the negative sign. For example, if we have the expression $-2(x^2 + 3x - 4)$, we can distribute the negative sign as follows:

$$-2(x^2 + 3x - 4) = -2x^2 - 6x + 8$$

Q: What is the difference between a quadratic expression and a linear expression?

A: A quadratic expression is an expression that involves a variable raised to the power of 2, while a linear expression is an expression that involves a variable raised to the power of 1. For example, $x^2 + 3x - 4$ is a quadratic expression because it involves the variable $x$ raised to the power of 2, while $2x + 3$ is a linear expression because it involves the variable $x$ raised to the power of 1.

Q: How do I factor an algebraic expression?

A: Factoring an algebraic expression involves expressing it as a product of simpler expressions. For example, if we have the expression $x^2 + 5x + 6$, we can factor it as follows:

$$x^2 + 5x + 6 = (x + 2)(x + 3)$$

Q: What is the greatest common factor (GCF) of two algebraic expressions?

A: The greatest common factor (GCF) of two algebraic expressions is the largest expression that divides both expressions without leaving a remainder. For example, if we have two expressions $x^2 + 3x - 4$ and $x^2 + 2x - 3$, the GCF of them is $x^2 + x - 1$.

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, we need to isolate the variable on one side of the equation. We can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, if we have the equation $x + 2 = 5$, we can solve it by subtracting 2 from both sides:

$$x + 2 = 5$$

$$x = 5 - 2$$

$$x = 3$$

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is an expression that involves a fraction of two polynomials, while a polynomial expression is an expression that involves a sum of terms with variables raised to non-negative integer powers. For example, $\frac{x^2 + 3x - 4}{x + 1}$ is a rational expression because it involves a fraction of two polynomials, while $x^2 + 3x - 4$ is a polynomial expression because it involves a sum of terms with variables raised to non-negative integer powers.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we need to factor the numerator and denominator, and then cancel out any common factors. For example, if we have the expression $\frac{x^2 + 3x - 4}{x + 1}$, we can simplify it as follows:

$$\frac{x^2 + 3x - 4}{x + 1} = \frac{(x + 2)(x - 2)}{x + 1}$$

$$= \frac{(x + 2)(x - 2)}{x + 1} \cdot \frac{x - 2}{x - 2}$$

$$= \frac{(x + 2)(x - 2)^2}{(x + 1)(x - 2)}$$

$$= x - 2$$