Difference of squares: Using quadrilaterals

To solve this problem, we start by recognizing that the expression for the area of the rectangle is given by the formula $(3x+4)(3x-4)$. This can be simplified using the difference of squares:

$(3x+4)(3x-4) = (3x)^2 - (4)^2 = 9x^2 - 16$.

The problem also provides the dimensions of the rectangle as 5 and 13. The area of the rectangle can therefore also be calculated as $5 \times 13 = 65$.

We set the two expressions for the area equal to each other to find $x$:

$9x^2 - 16 = 65$.

Next, we solve for $x$:

$9x^2 - 16 = 65$
$9x^2 = 65 + 16$
$9x^2 = 81$
$x^2 = \frac{81}{9}$
$x^2 = 9$
$x = \pm 3$.

Therefore, the value of $x$ is $\operatorname{\pm}3$.

To find $x$, we'll begin by calculating the area of the rectangle using its dimensions:

• Step 1: Calculate the area $A$ using the dimensions 5 and 19:
  $A = 5 \times 19 = 95$
• Step 2: Set up the equation using the difference of squares:
  $(x-7)(x+7) = x^2 - 49$
• Step 3: Equate to the area calculated:
  $x^2 - 49 = 95$
• Step 4: Solve for $x^2$:
  $x^2 = 95 + 49 = 144$
• Step 5: Solve for $x$ by taking the square root:
  $x = \pm \sqrt{144} = \pm 12$

Therefore, the solutions for $x$ are $x = \pm 12$.

The problem requires calculating the value of $x$ from the expression given for the area of a parallelogram: $(x+3)(x-3)$.

Recognize that the expression $(x+3)(x-3)$ can be expanded using the identity for the difference of squares:

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

Thus, it simplifies to:

$x^2 - 9$

Understanding from the problem that this represents the area of the parallelogram, and after setting it equal to zero:

$x^2 - 9 = 0$

To solve for $x$, add 9 to both sides to isolate $x^2$:

$x^2 = 9$

Take the square root of both sides, remembering that squaring gives two solutions:

$x = \pm \sqrt{9} = \pm 3$

Thus, the solutions for $x$ are $x = 3$ and $x = -3$.

Therefore, the value of $x$ is $\pm6$.

Therefore, the solution to the problem is $x = \pm3$.

To solve the problem, let's proceed with the following steps:

• Step 1: Identify the area formula
• Step 2: Substitute side expressions
• Step 3: Simplify the equation and solve

Let's begin:
Step 1: The area of the rectangle is given by the formula:
$\text{Area} = \text{Length} \times \text{Width}$ For this rectangle, the length and width are given as $2x + 8$ and $2(x-4)$, respectively. Thus, we have the equation for the area:

$(2x + 8) \times (2(x - 4)) = 132$

Step 2: Substitute the side expressions and set up the equation:

$(2x + 8) \times (2x - 8) = 132$

Step 3: Expand and solve:

• First, we expand the expression using multiplication:
• The side expression simplifies to a difference of squares:

$(2x + 8)(2x - 8) = (2x)^2 - 8^2$ $= 4x^2 - 64$

The area equation is:

$4x^2 - 64 = 132$

Step 4: Simplify and solve the quadratic equation:

• Add $64$ to both sides to isolate the quadratic term:
$4x^2 = 196$
• Divide by $4$:
$x^2 = 49$
• Take the square root:
$x = \pm \sqrt{49}$ $x = \pm 7$

Therefore, the solutions for $x$ are $x = 7$ and $x = -7$. Since area is scalar, both positive and negative solutions are valid for $x$.

Thus, the correct value for $x$ is $\pm 7$.

To solve this problem, we'll calculate $x$ using the provided expressions for the base and height of the parallelogram.

Given the area of the parallelogram:

$A = (\text{base}) \times (\text{height})$

In our case, the base is $x + 5$, and the height is $x - 5$. Therefore, we have:

$(x + 5)(x - 5) = 56$

Recognizing this as a difference of squares, we write:

$x^2 - 25 = 56$

Add 25 to both sides to isolate $x^2$:

$x^2 = 81$

Take the square root of both sides:

$x = \pm 9$

Since both dimensions of a parallelogram must be positive in practical applications, we take $x = 9$.

Therefore, the correct solution is $x = 9$.

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

• Step 1: Simplify the expression $(4x+9)(4x-9)$ using the difference of squares formula.
• Step 2: Calculate the area using the rectangle's given dimensions, 9 and 7.
• Step 3: Equate the expressions and solve for $x$.

Step 1: Simplifying $(4x+9)(4x-9)$.
This expression is a difference of squares, so it simplifies as follows:

$(4x+9)(4x-9) = (4x)^2 - 9^2 = 16x^2 - 81.$

Step 2: Calculate the area using given rectangle dimensions: 9 and 7.
The area $A$ of the rectangle is:

$A = 9 \times 7 = 63.$

Step 3: Equate the two expressions:
$16x^2 - 81 = 63.$
To solve for $x$, start by adding 81 to both sides:

$16x^2 = 144.$

Divide both sides by 16 to isolate $x^2$:

$x^2 = 9.$

Take the square root of both sides to solve for $x$:

$x = \pm\sqrt{9} = \pm3.$

Therefore, the solution to the problem is $x = \pm3$.