Problem 73 - Elementary Algebra

Question

The sides of a square measure \(x+3\) units. If the area is 25 square units, then find \(x\).

Step-by-Step Solution

Answer

The value of \(x\) is 2.

Step 1: Write the area formula for a square

The area of a square is calculated by squaring the length of one of its sides. Since the side of the square is given as \(x+3\), we can write the area as \((x+3)^2\).

Step 2: Set up the equation for area

We know the area is given as 25 square units. Therefore, we set the equation as: \((x+3)^2 = 25\).

Step 3: Solve the equation

To solve \((x+3)^2 = 25\), we take the square root of both sides to get \(x+3 = \pm 5\). This gives us two equations to solve: \(x+3 = 5\) and \(x+3 = -5\).

Step 4: Solve for x

First equation: Solve \(x+3 = 5\) by subtracting 3 from both sides, resulting in \(x = 2\). Second equation: Solve \(x+3 = -5\) similarly to get \(x = -8\).

Step 5: Validate the solutions

The side length of a square is always positive, so \(x+3\) must be positive. Thus, \(x+3 = -5\) is not valid. Therefore, \(x = 2\) is the only valid solution.

Key Concepts

Area of a Square

The concept of the "area of a square" is fundamental in geometry and mathematics. The area is defined as the space contained within the boundaries of the square. For any square, the area calculation is straightforward: you square the length of one side.

This is because a square has all four sides of equal length. So, if each side is of length 's', the area becomes \( s^2 \).

• For example, if a square’s side is 4 units, the area is \( 4 \times 4 = 16 \) square units.
• If the side length changes, the area changes proportionally.

This concept is applied to solve problems where side lengths are given in terms of variables, such as \( x+3 \) in the problem above.

By understanding that the area is \( (x+3)^2 \), we're using the standard area formula adapted to variables, which is often necessary in algebraic contexts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They represent quantities without being directly solved until values are substituted or further conditions are applied.

In the context of this exercise, the expression \( (x+3) \) represents the length of the square's side. To determine the area of the square, we create the expression \( (x+3)^2 \). This is a perfect example of how algebraic expressions are used to transform a geometric problem into an algebraic equation.

Algebraic expressions can often be playful or complex, involving:

• Operations like addition, subtraction, multiplication, and division.
• Exponents as seen in squaring terms \((x+3)^2\).
• Combining like terms and simplifying.

In our problem, knowing to apply the square root operation to an algebraic expression like \( (x+3)^2 \) equals 25 helps us isolate the variable \( x \) to find its specific value.

Validating Solutions

Validating solutions is a crucial part of solving any mathematical problem. It ensures that the solution satisfies all conditions originally posed by the problem.

In this exercise, after obtaining solutions for \( x \) through factors of the original equation, both solutions \( x = 2 \) and \( x = -8 \) needed to be validated. It's important to remember the context:

For geometric problems, such as squares, only positive numbers are considered realistic lengths. So, even if the mathematics provide a negative solution:

• We must discard any side length calculations resulting in a negative value since length cannot be negative.
• Ensure logical consistency, meaning only \( x = 2 \) fits as it results in a valid (positive) side length.

By validating, you confirm that the solution is complete and that any extraneous solutions are properly discarded, ensuring not just a mathematical correctness, but practical applicability as well. This step is as significant as any computational process in problem-solving.