Problem 49 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (9 x+8)(9 x-8)=81 x^{2}-64 $$

Step-by-Step Solution

Answer

The statement is true, so there is no need to rewrite the right-hand side.

Step 1: Recognize the difference of squares

Identify the equation given, \((9x + 8)(9x - 8) = 81x^{2} - 64\), as the difference of squares, with a = \(9x\) and b = 8.

Step 2: Apply the difference of squares property

The difference of squares property is that \(a^2 - b^2 = (a + b)(a - b)\). In this case, a is \(9x\) and b is 8. Applying this property, the right side of the equation becomes \((9x)^2 - (8)^2\). Cubing these terms, we get \(81x^2 - 64\).

Step 3: Compare with the original statement

The right-hand side of the equation matches with the original statement given, \(81x^2 - 64\). So, the statement is true.

Key Concepts

Algebraic expressions

Algebraic expressions are combinations of variables and constants. In our problem, the expression is \((9x + 8)(9x - 8)\). **Understanding Algebraic Expressions:** These expressions can include variables like \(x\), numbers like 9 and 8, and operators such as addition and multiplication. The **goal is to simplify or solve these expressions.**

A simplified algebraic expression is when you cannot combine any more like terms. For example, combining like terms in simple expressions such as \(2x + 3x\) to get \(5x\). Algebraic expressions are the building blocks of more complex algebraic equations.

**Importance of Parentheses:** In the given expression \((9x + 8)(9x - 8)\), parentheses indicate multiplication between the two binomials.

• The expression inside each parenthesis is treated as a single entity.
• Ensures the correct application of operations when dealing with multiple terms.

Simplifying algebraic expressions properly involves knowing operations and the rules of arithmetic, such as distribution and combining like terms.

Equation solving

Equation solving involves finding the value of the variables that make an equation true. Let's consider our example equation: \((9x + 8)(9x - 8) = 81x^2 - 64\). **Solving Equations with Algebraic Expressions:**

To solve equations, you often have to simplify both sides first. In this problem, the left side is already expanded into a form that represents the difference of squares. Solving this means checking if both sides of the equation are equivalent.

**Steps to Solve an Equation:**

• Combine like terms.
• Use mathematical properties, such as the difference of squares, to simplify complex sections.
• Check if both sides equal by simplifying both sides to see if they produce the same expression.

In our case, by looking at both sides and using the difference of squares property, we find that they match, showing us that the equation is true.

Mathematical properties

Mathematical properties are fundamental rules that govern how numbers interact in equations. The exercise focuses on the difference of squares, which is one of these properties. **Understanding the Difference of Squares:** This property is given by the formula \(a^2 - b^2 = (a + b)(a - b)\). This is crucial for simplifying expressions like the one in our problem.

**How to Apply Difference of Squares:**
The formula helps factor expressions efficiently and identify a specific pattern:

• If you recognize the format \(a^2 - b^2\), you can directly break it down into \((a + b)(a - b)\).
• This pattern saves time in solving complex equations and simplifying expressions.
• In our solution: \((9x)^2 - 8^2\) directly forms \((9x + 8)(9x - 8)\).

Understanding this property helps in improving problem-solving skills as it reduces the effort needed to manipulate and solve algebraic expressions.