Calculating the Area of a Rectangle in Terms of x: A Step-by-Step Guide

Understanding the area of a rectangle is a fundamental concept in geometry and algebra. The area of a rectangle can be calculated using the simple formula:

Area Length × Width

Given a rectangle with a length of (x - 6) meters and a width of (x - 2) meters, we can express the area in terms of (x). In this article, we will guide you through the steps to find the area in terms of (x) and explore the implications of the resulting expression.

Step-by-Step Solution

To find the area, we need to multiply the length and the width:

Length (x - 6) Width (x - 2)

The area (A) can thus be expressed as:

[text{Area} (x - 6)(x - 2)]

Expanding this expression using the distributive property (also known as the FOIL method for binomials), we get:

[(x - 6)(x - 2) x(x - 2) - 6(x - 2)]

Now, let's distribute (x) and (-6):

[(x - 6)(x - 2) x^2 - 2x - 6x 12]

Combining like terms, we have:

[(x - 6)(x - 2) x^2 - 8x 12]

Therefore, the area (A) of the rectangle in terms of (x) is:

[text{Area} x^2 - 8x 12 text{ square meters}]

Applying the Quadratic Formula

We can also solve this problem using the quadratic formula. Consider another scenario where we are given:

[text{Area} (x - 5)(x - 3) s^2 - 2x - 15]

Expanding the left side, we get:

[(x - 5)(x - 3) x^2 - 3x - 5x 15 x^2 - 8x 15]

Equating this to the given area, we have:

[text{Area} x^2 - 8x 15 s^2 - 2x - 15]

Rewriting it, we get:

[text{Area} s^2 - 2x - 15]

We can solve for (s) using the quadratic formula. Given the equation:

[(x - 5)(x - 3) s^2 - 2x - 15]

The quadratic formula (x frac{-b pm sqrt{b^2 - 4ac}}{2a}) can help us solve for (x). For the equation (x^2 - 8x 15 s^2 - 2x - 15), we can solve for (s^2). Simplifying, we get:

[s^2 - 2x - 15 x^2 - 8x 15]

Thus:

[s^2 x^2 - 6x 30]

We can then solve for (s) using the quadratic formula:

[s sqrt{x^2 - 6x 30}]

Interpreting the Results

For (x 3), the area is calculated as:

[(3 - 5)(3 - 3) (3 - 5)(0) 0]

For (x -5), the area is:

[(x - 5)(x - 3) (-5 - 5)(-5 - 3) (-10)(-8) 80]

When (x 3), the area is zero, indicating that the dimensions do not form a valid rectangle. For (x -5), the area is 80 square meters, which is valid.

Conclusion

Remember, the area of a rectangle in terms of (x) is given by:

[text{Area} x^2 - 8x 12]

In conclusion, the area of the rectangle in terms of (x) is a key concept in algebra and geometry. Understanding how to manipulate and solve algebraic expressions can help in various real-world applications, from construction to design. By following the steps outlined in this guide, you can effectively calculate and interpret the area of a rectangle in terms of (x).