Square of sum: Using short multiplication formulas

To find the area of a square with side length $x + 1$, we apply the formula for the area of a square, which is side squared. This means we need to calculate $(x + 1)^2$.

Here are the steps to solve the problem:

• Step 1: Identify the expression for the side length. The side length of the square is given as $x + 1$.
• Step 2: Use the formula for the area of a square: $(\text{side})^2$.
• Step 3: Substitute the side length with $x + 1$: $(x + 1)^2$.
• Step 4: Expand the expression using the formula for the square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$, where $a = x$ and $b = 1$.
• Step 5: Calculation:
• $(x + 1)^2 = x^2 + 2(x)(1) + 1^2$
• $(x + 1)^2 = x^2 + 2x + 1$

Therefore, the algebraic expression for the area of the square is $x^2 + 2x + 1$.

To solve this problem, follow these steps:

• Step 1: Set up the equation based on the given information. The side of the square is given by $x + 1$, and the area formula for a square is $(\text{side})^2$. Thus, we set up the equation as $(x + 1)^2 = 36$.
• Step 2: Solve for $x$. Take the square root of both sides of the equation:

$(x + 1) = \pm \sqrt{36}$
$(x + 1) = \pm 6$

• Since $x > 0$ is a condition, only the positive value of $x + 1$ is valid, so $x + 1 = 6$.
• Solve for $x$ by subtracting 1 from both sides:

$x = 6 - 1$
$x = 5$

Therefore, the solution to the problem is $x = 5$.