Solve for X: Area vs. Perimeter in Two Altered Square Equations

To find the length of the smaller square, we need to solve the equation derived from the problem statement:

• Step 1: Write down the equation for the area of the larger square: $(x + 2)^2$.
• Step 2: Write down the equation for the perimeter of the smaller square: $4x$.
• Step 3: Set up the equation as given: $(x + 2)^2 = 4x + 20$.

Let's solve the equation:

Step 1: Expand $(x + 2)^2$:

$(x + 2)^2 = x^2 + 4x + 4$

Step 2: Rewrite the equation substituting the expanded form:

$x^2 + 4x + 4 = 4x + 20$

Step 3: Simplify by eliminating $4x$ from both sides:

$x^2 + 4 = 20$

Step 4: Subtract 4 from both sides:

$x^2 = 16$

Step 5: Take the square root of both sides:

$x = 4$ or $x = -4$

Since $x$ must be positive, we have:

$x = 4$

Thus, the length of the side of the smaller square is $4$.