Solve for X: Understanding the Rectangle's Area via (4x+9)(4x-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 $x = \pm3$.