Difference of squares: Worded problems

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

With this is mind, let's proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$

$$We are told that one side of the rectangle is obtained by extending one side of the square with side length $x$ (cm) by 3 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 3 cm:

$$

Therefore, the lengths of the rectangle's sides are:

$EF=HG=x+3\\ EH=FG=x-3$cm,

Apply the above formula in order to calculate the area of the rectangle that was formed from the square as described in the question:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3)$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the difference of squares formula:

$(c+d)(c-d)=c^2-d^2$The area of the rectangle using the above formula is as follows:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3^2\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-9}$(sq cm)

Therefore, the correct answer is answer C.

To solve this problem, we'll follow these steps:

• Step 1: Identify the areas for both square and rectangle using given dimensions.
• Step 2: Equate the two areas as the problem states they are equal.
• Step 3: Solve the resulting equation for $x$.
• Step 4: Check that the condition $x > 1$ is satisfied and select the correct answer choice.

Now, let's work through the solution:

Step 1: The area of the square with side length $x$ is given by:

$x^2$.

For the rectangle, where one side is extended by 3 cm and an adjacent side is shortened by 1 cm, we have:

Original length and width of the rectangle are $x + 3$ and $x - 1$, respectively.

The area of the rectangle becomes:

$(x + 3)(x - 1)$.

Step 2: As per the problem, these two areas are equal:

$x^2 = (x + 3)(x - 1)$.

Step 3: Expanding the right-hand side of the equation:

$(x + 3)(x - 1) = x^2 + 3x - x - 3 = x^2 + 2x - 3$.

Now, equate and simplify:

$x^2 = x^2 + 2x - 3$.

Subtract $x^2$ from both sides:

$0 = 2x - 3$.

Adding 3 to both sides gives:

$3 = 2x$.

Divide both sides by 2 to solve for $x$:

$x = \frac{3}{2}$.

Step 4: We check the condition $x > 1$ and find $x = \frac{3}{2} \approx 1.5$ which satisfies it.

Therefore, the side length of the square is $x = \frac{3}{2}$ cm.

First, recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

Proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$ (drawing)

$$It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x +1$ (cm) by 1 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 1 cm:

Therefore, the lengths of the rectangle's sides are:

$EF=HG=(x+1)+1\\ \downarrow\\ \boxed{ EF=HG=x+2}\\ \hspace{2pt}\\ \\ EH=FG=(x+1)-1\\ \downarrow\\ \boxed{ EH=FG=x }$ (cm)

Apply the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the distributive property:

$(m+n)d=md+nd$ Therefore, applying the distributive property, we obtain the following area for the rectangle:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x \\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+2x}$(sq cm)

The correct answer is answer B.

To determine which shape has a larger area, we need to compare the areas of the square and the rectangle:

• Step 1: Calculate the area of the original square:

The side length of the square is $X$, so its area is given by:

• Step 2: Calculate the area of the rectangle:

The dimensions of the rectangle are $X + 3$ cm and $X - 3$ cm. Thus, its area is:

Using the difference of squares formula, we find:

• Step 3: Compare the areas:

We compute the difference between the square's area and the rectangle's area:

Since 9 is positive, the area of the square is larger than the area of the rectangle.

Therefore, the square has a larger area than the rectangle.