Problem 359 - Elementary Algebra

Question

Area of a patio If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation \((s+4)^{2}=196\) for s to find the length of a side of the patio.

Step-by-Step Solution

Answer

The length of a side of the patio is 10 feet.

Step 1 - Understand the Problem

We are given that the side length of a square patio is increased by 4 feet, resulting in an area of 196 square feet. We need to find the original length of a side of the patio, which is represented by the variable \(s\).

Step 2 - Write the Given Equation

The problem gives us the equation \( (s + 4)^2 = 196 \) where \(s\) is the original side length of the patio.

Step 3 - Take the Square Root of Both Sides

To solve for \(s\), first we need to take the square root of both sides of the equation: \[(s + 4)^2 = 196\] \[s + 4 = \sqrt{196}\] \[s + 4 = 14\]

Step 4 - Solve for \(s\)

Isolate \(s\) by subtracting 4 from both sides of the equation: \[s + 4 - 4 = 14 - 4\] \[s = 10\]

Step 5 - Verify the Solution

Double-check by substituting \(s = 10\) back into the original problem to make sure it fits: \( (10 + 4)^2 = 14^2 = 196 \). This confirms the solution is correct.

Key Concepts

Area of a Square

The exercise involves understanding the concept of the area of a square. The area of a square is calculated by squaring the length of one of its sides. Mathematically, this can be expressed as:
\[ \text{Area} = s^2 \] where \(s\) is the length of one side of the square.
In the given problem, if each side of the square patio is increased by 4 feet, the new side length becomes \(s + 4\). The problem states that this new side length results in an area of 196 square feet. We can write this as:
\[ (s + 4)^2 = 196 \] To solve this, we will use algebraic methods.
By understanding how the area of a square is related to its side length, we can set up and solve the given equation.

Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics. In this exercise, the equation given is \((s + 4)^2 = 196\). This is a quadratic equation, which can often be solved by taking the square root of both sides.
Here are the steps to solve it:

• First, take the square root of both sides:
  \[ \text{Equation:} \ (s + 4)^2 = 196 \ \text{Square Root of Both Sides:} \ s + 4 = \sqrt{196} \]
  
• Next, simplify the square root: \[s + 4 = 14 \]
  
• Finally, to isolate \(s\), subtract 4 from both sides: \[s + 4 - 4 = 14 - 4 \ s = 10 \]

As you can see, algebra helps us systematically find the values of variables in equations.
Double-check your solutions by substituting the value back into the original equation.
For our case: \((10 + 4)^2\) do indeed equals \(196\). This verifies our solution is correct.

Square Roots

Understanding square roots is essential for solving quadratic equations. A square root of a number \(x\) is another number \(y\) such that \(y^2 = x\). It is denoted as \(\sqrt{x}\).
In the context of our exercise, we need to find the square root of 196. By definition: \[ \sqrt{196} = 14 \]
This tells us that 14 multiplied by itself (14 * 14) equals 196.
Keep in mind that every positive number has both a positive and a negative square root. However, for the purpose of this problem—dealing with physical lengths—we only consider the positive square root.
Once we found that \(s + 4 = 14\), we can isolate \(s\) to find the original length of the patio. Taking square roots helps simplify and solve quadratic equations by breaking them down into more straightforward linear equations.