Simplify The Expression: ( X − 4 B ) − 2 ( 3 B + X ) 2 (x - 4b) - 2(3b + X)^2 ( X − 4 B ) − 2 ( 3 B + X ) 2

Mar 13, 2025 by ADMIN 108 views

**Simplify the Expression: $(x - 4b) - 2(3b + x)^2$**

Introduction

In this article, we will simplify the given expression $(x - 4b) - 2(3b + x)^2$ . This involves expanding the squared term, distributing the negative sign, and combining like terms. We will use the order of operations (PEMDAS) to ensure that we simplify the expression correctly.

Step 1: Expand the Squared Term

The first step is to expand the squared term $(3b + x)^2$ . To do this, we will use the formula $(a + b)^2 = a^2 + 2ab + b^2$ . In this case, $a = 3b$ and $b = x$ .

(3b + x)^2 = (3b)^2 + 2(3b)(x) + x^2

Expanding the squared term, we get:

(3b + x)^2 = 9b^2 + 6bx + x^2

Step 2: Distribute the Negative Sign

The next step is to distribute the negative sign to the terms inside the parentheses. This means that we will multiply each term inside the parentheses by $-1$ .

(x - 4b) - 2(3b + x)^2 = (x - 4b) - 2(9b^2 + 6bx + x^2)

Distributing the negative sign, we get:

(x - 4b) - 2(3b + x)^2 = x - 4b - 18b^2 - 12bx - 2x^2

Step 3: Combine Like Terms

The final step is to combine like terms. This means that we will add or subtract terms that have the same variable and exponent.

x - 4b - 18b^2 - 12bx - 2x^2 = -18b^2 - 12bx - 2x^2 + x - 4b

Combining like terms, we get:

x - 4b - 18b^2 - 12bx - 2x^2 = -18b^2 - 11bx - 2x^2 - 4b

Conclusion

In this article, we simplified the expression $(x - 4b) - 2(3b + x)^2$ . We expanded the squared term, distributed the negative sign, and combined like terms. The final simplified expression is $-18b^2 - 11bx - 2x^2 - 4b$ .

Tips and Tricks

When simplifying expressions, it's essential to follow the order of operations (PEMDAS).
Use the distributive property to expand and simplify expressions.
Combine like terms to simplify expressions.

Common Mistakes

Failing to follow the order of operations (PEMDAS).
Not distributing the negative sign correctly.
Not combining like terms correctly.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics and has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems.

Practice Problems

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

References

[1] "Algebra" by Michael Artin.
[2] "Calculus" by Michael Spivak.
[3] "Mathematics for Computer Science" by Eric Lehman.

Introduction

In our previous article, we simplified the expression $(x - 4b) - 2(3b + x)^2$ . We expanded the squared term, distributed the negative sign, and combined like terms. In this article, we will answer some common questions related to simplifying expressions.

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 simplifying expressions. PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
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 the negative sign?

A: To distribute the negative sign, you need to multiply each term inside the parentheses by $-1$ . For example, if you have the expression $(x - 4b) - 2(3b + x)^2$ , you would distribute the negative sign as follows:

(x - 4b) - 2(3b + x)^2 = (x - 4b) - 2(9b^2 + 6bx + x^2)

Distributing the negative sign, you get:

(x - 4b) - 2(3b + x)^2 = x - 4b - 18b^2 - 12bx - 2x^2

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract terms that have the same variable and exponent. For example, if you have the expression $-18b^2 - 12bx - 2x^2 + x - 4b$ , you can combine like terms as follows:

-18b^2 - 12bx - 2x^2 + x - 4b = -18b^2 - 11bx - 2x^2 - 4b

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Failing to follow the order of operations (PEMDAS)
Not distributing the negative sign correctly
Not combining like terms correctly

Q: How do I apply simplifying expressions in real-world scenarios?

A: Simplifying expressions is a crucial skill in mathematics and has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems.

Q: What are some practice problems to help me improve my skills?

A: Here are some practice problems to help you improve your skills:

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

Conclusion

In this article, we answered some common questions related to simplifying expressions. We discussed the order of operations (PEMDAS), distributing the negative sign, combining like terms, and common mistakes to avoid. We also provided some practice problems to help you improve your skills.

Tips and Tricks

When simplifying expressions, it's essential to follow the order of operations (PEMDAS).
Use the distributive property to expand and simplify expressions.
Combine like terms to simplify expressions.

Real-World Applications

Practice Problems

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

References

[1] "Algebra" by Michael Artin.
[2] "Calculus" by Michael Spivak.
[3] "Mathematics for Computer Science" by Eric Lehman.

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.