Problem 101 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (3 c+10)(3 c-10) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \((3c + 10)(3c - 10)\). Answer: The simplified algebraic expression is \(9c^2 - 100\).

Step 1: Identify a and b

In the given expression, \((3c + 10)(3c - 10)\), we can see that it is in the form \((a + b)(a - b)\). Comparing the given expression with this form, we can identify \(a = 3c\) and \(b = 10\).

Step 2: Apply the difference of squares formula

Now we can apply the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). Substitute the values of \(a\) and \(b\) we found in step 1: $$(3c + 10)(3c - 10) = (3c)^2 - (10)^2$$

Step 3: Simplify the expression

Finally, we can simplify the expression by squaring the terms \(3c\) and \(10\): $$(3c)^2 - (10)^2 = 9c^2 - 100$$ The simplified algebraic expression is: $$9c^2 - 100$$

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables and constants connected by various mathematical operations such as addition, subtraction, multiplication, or division. In our example with \((3c + 10)(3c - 10)\), the expression contains variables (\(c\) in this case) and constants, numbers that do not change. An important part of working with algebraic expressions is the identification of these components and understanding their role in the expression.
By breaking down these expressions into their individual components, you can better understand how they function and interact. This is especially useful when simplifying expressions or solving equations. When you see terms grouped together in parentheses like \((3c + 10)\), these terms are meant to be treated as a single entity in operations like multiplication or factoring.

The Art of Simplification

Simplification in algebra is a process of making an expression more compact and manageable. It often involves combining like terms, using distributive properties, or employing known formulas to reduce the complexity of an equation or expression.
In our problem, the expression \((3c + 10)(3c - 10)\) is simplified by recognizing it as a "difference of squares." This term describes a common algebraic pattern expressed as \((a + b)(a - b)\), which simplifies to \(a^2 - b^2\).

• Recognize patterns or terms that can be factored or expanded using known identities or formulas.
• Simplify by performing arithmetic operations: for (\(3c\))², you multiply 3 by itself, giving you \(9c^2\), and for 10², you multiply 10 by itself, resulting in 100.

The goal of simplification is to make the expression as straightforward as possible, which can help solve the equations or understand the problem effectively.

The Power of Mathematical Formulas

Mathematical formulas serve as shortcuts for performing complex calculations. They embody common patterns and hold the key to unlocking complex expressions. Recognizing and applying the correct formula can simplify tasks and deepen understanding.
In algebra, the formula for the difference of squares is particularly useful. It's stated as \(a^2 - b^2 = (a + b)(a - b)\). This formula allowed us to simplify the expression \((3c + 10)(3c - 10)\) quickly and efficiently.

• Understanding how and when to apply specific formulas can drastically reduce the time needed to solve algebraic problems.
• Formulas help us see the inherent relationships and symmetry within mathematical expressions, making them easier to handle.

By mastering these formulas, students can tackle problems more effectively and develop a deeper appreciation for the structure and efficiency that mathematics provides.