Distribute the negative sign

Oct 13, 2025 by talynt.org 5 views

Introduction

Algebraic operations are a fundamental aspect of mathematics, and understanding how to perform them is crucial for solving equations and manipulating expressions. In this article, we will focus on performing three different algebraic operations: subtraction, multiplication, and squaring. We will break down each operation step-by-step and provide examples to illustrate the concepts.

Subtraction Operation (a)

The first operation we will perform is the subtraction of two expressions: $(8m - 4) - (8m + 4)$ . To do this, we need to follow the order of operations (PEMDAS):

Distribute the negative sign: When we see a negative sign in front of a parentheses, we need to distribute it to each term inside the parentheses. In this case, we have:

$-(8m + 4) = -8m - 4$

Combine like terms: Now that we have distributed the negative sign, we can combine like terms. In this case, we have:

$(8m - 4) - (8m + 4) = 8m - 4 - 8m - 4$

Simplify the expression: Finally, we can simplify the expression by combining like terms:

$8m - 4 - 8m - 4 = -8$

Therefore, the result of the subtraction operation is $-8$ .

Multiplication Operation (b)

The second operation we will perform is the multiplication of two expressions: $(8m - 4)(8m + 4)$ . To do this, we need to follow the distributive property:

Distribute the first term: We will multiply the first term of the first expression ( $8m$ ) with each term of the second expression:

$8m(8m) + 8m(4)$

Distribute the second term: Next, we will multiply the second term of the first expression ( $-4$ ) with each term of the second expression:

$-4(8m) - 4(4)$

Combine like terms: Now that we have distributed each term, we can combine like terms:

$64m^2 + 32m - 32m - 16$

Simplify the expression: Finally, we can simplify the expression by combining like terms:

$64m^2 - 16$

Therefore, the result of the multiplication operation is $64m^2 - 16$ .

Squaring Operation (c)

The third operation we will perform is the squaring of an expression: $(8m - 4)^2$ . To do this, we need to follow the formula for squaring a binomial:

Square the first term: We will square the first term of the expression ( $8m$ ):

$(8m)^2 = 64m^2$

Square the second term: Next, we will square the second term of the expression ( $-4$ ):

$(-4)^2 = 16$

Multiply the terms: Now that we have squared each term, we can multiply them together:

$64m^2 \cdot 16 = 1024m^2$

Add the terms: Finally, we can add the terms together:

$1024m^2 + 2 \cdot 8m \cdot (-4) + (-4)^2$

Simplify the expression: We can simplify the expression by combining like terms:

$1024m^2 - 64m - 16$

Therefore, the result of the squaring operation is $1024m^2 - 64m - 16$ .

Conclusion

In this article, we have performed three different algebraic operations: subtraction, multiplication, and squaring. We have broken down each operation step-by-step and provided examples to illustrate the concepts. By following the order of operations and using the distributive property, we can simplify complex expressions and solve equations. With practice and patience, anyone can master these algebraic operations and become proficient in mathematics.

Key Takeaways

To perform subtraction, we need to follow the order of operations (PEMDAS) and distribute the negative sign.
To perform multiplication, we need to follow the distributive property and multiply each term of one expression with each term of the other expression.
To perform squaring, we need to follow the formula for squaring a binomial and multiply the terms together.

Practice Problems

Perform the subtraction operation: $(3x + 2) - (3x - 2)$
Perform the multiplication operation: $(2x + 1)(3x - 1)$
Perform the squaring operation: $(x + 2)^2$

Answer Key

$6$
$6x^2 - 2x - 3$
$x^2 + 4x + 4$
Algebraic Operations Q&A ==========================

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next (e.g., 2^3).
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: When we see a negative sign in front of a parentheses, we need to distribute it to each term inside the parentheses. For example:

$-(2x + 3) = -2x - 3$

Q: What is the distributive property?

A: The distributive property is a rule that allows us to multiply a single term to multiple terms inside parentheses. For example:

$(2x + 3)(4) = 2x(4) + 3(4) = 8x + 12$

Q: How do I square a binomial?

A: To square a binomial, we can use the formula:

$(a + b)^2 = a^2 + 2ab + b^2$

For example:

$(x + 2)^2 = x^2 + 2x(2) + 2^2 = x^2 + 4x + 4$

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example:

$x$ is a variable

A constant is a value that does not change. For example:

$2$ is a constant

Q: How do I simplify an expression?

A: To simplify an expression, we need to combine like terms and eliminate any unnecessary parentheses. For example:

$2x + 3x = 5x$

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two expressions are equal. For example:

$2x + 3 = 5$

An expression is a group of terms that are combined using mathematical operations. For example:

$2x + 3$

Q: How do I solve an equation?

A: To solve an 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:

$2x + 3 = 5$

Subtracting 3 from both sides:

$2x = 2$

Dividing both sides by 2:

$x = 1$

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

A: A linear equation is an equation that can be written in the form:

$ax + b = c$

For example:

$2x + 3 = 5$

A quadratic equation is an equation that can be written in the form:

$ax^2 + bx + c = 0$

For example:

$x^2 + 4x + 4 = 0$

Q: How do I graph a linear equation?

A: To graph a linear equation, we need to find two points on the line and plot them on a coordinate plane. We can then draw a line through the two points to represent the equation. For example:

$y = 2x + 3$

We can find two points on the line by plugging in different values of x and solving for y. For example:

When x = 0, y = 3
When x = 1, y = 5

We can then plot these two points on a coordinate plane and draw a line through them to represent the equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, we need to find the vertex of the parabola and plot it on a coordinate plane. We can then draw a parabola through the vertex to represent the equation. For example:

$x^2 + 4x + 4 = 0$

We can find the vertex of the parabola by using the formula:

$x = -\frac{b}{2a}$

In this case, a = 1 and b = 4, so:

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

We can then plot the vertex on a coordinate plane and draw a parabola through it to represent the equation.

Conclusion

In this article, we have answered some of the most frequently asked questions about algebraic operations. We have covered topics such as the order of operations, distributing negative signs, the distributive property, squaring binomials, and simplifying expressions. We have also discussed the difference between variables and constants, equations and expressions, and linear and quadratic equations. Finally, we have provided some tips for graphing linear and quadratic equations. With practice and patience, anyone can master these algebraic operations and become proficient in mathematics.