Solving the Rectangle Problem: Finding Length and Width Using Quadratic Equations

Understanding the principles of quadratic equations is essential for solving geometric problems. One such problem is finding the length and width of a rectangle when the area and a relationship between the length and width are given. In this article, we will walk through the process of solving such a problem step by step.

Problem Statement

The area of a rectangle is 56 square centimeters. If the length is 2 more than 3 times the width, what are the length and the width?

Formulation of the Equation

Let's denote the width of the rectangle by x (in cm). Given that the length is 2 more than 3 times the width, we can express the length as 3x 2 cm. The area of the rectangle is given as 56 square centimeters, which can be represented by the equation:

Area Length × Width

Substituting the given values, we get:

(3x 2) × x 56

Solving the Quadratic Equation

To solve this equation, we first expand it:

3x^2 2x 56

Next, we rearrange the equation to a standard quadratic form by moving all terms to one side:

3x^2 2x - 56 0

Now, we solve this quadratic equation using the factoring method. We need to find two numbers that multiply to give -168 and add to give 2 (the coefficient of the x term). After some trial and error, we find that these numbers are -12 and 14. We can then factor the equation as follows:

(3x - 12)(x 4) 0

This gives us two solutions for x:

3x - 12 0 or x 4 0

Solving these equations:

x 4 (since x cannot be negative in this context)

Thus, the width of the rectangle is 4 cm.

Using the relationship between the length and the width, we can now find the length:

Length 3x 2 3(4) 2 12 2 14 cm

Verification

To verify our solution, we calculate the area using the found dimensions:

Area 4 cm × 14 cm 56 cm^2

The calculated area matches the given area, confirming our solution is correct.

Conclusion

In this article, we have tackled the problem of finding the length and width of a rectangle given the area and a relationship between the length and width. By formulating the problem into a quadratic equation and solving it using the factoring method, we found that the width of the rectangle is 4 cm and the length is 14 cm. The process is a practical application of solving quadratic equations in real-life scenarios.