Solving Quadratic Equations in Practical Applications: The Dimensions of a Bedroom
Quadratic equations are a fundamental concept in mathematics, widely used in various real-world applications, including geometry. This article explores how to solve a practical problem using quadratic equations, specifically finding the dimensions of a bedroom. By the end of this article, you'll have a clear understanding of the process and be able to apply similar techniques to other similar problems.
Problem Statement
Imagine you are tasked with determining the dimensions of a rectangular bedroom given certain constraints. The width of the bedroom is 5 meters less than its length, and its area is 84 square meters. The goal is to determine the exact dimensions of the bedroom.
Step-by-Step Solution
Let's denote the length of the bedroom as L meters. Therefore, the width can be expressed as L - 5 meters. The area of the bedroom is given as 84 square meters, which can be mathematically represented as:
Area Length x Width L(L - 5) 84
This equation simplifies to:
L2 - 5L 84
To solve this equation, we first rearrange it into standard quadratic form:
L2 - 5L - 84 0
We can solve this quadratic equation using the quadratic formula:
L [-b ± (b2 - 4ac)?]/2a
Here, the values of the coefficients are: a 1, b -5, and c -84. Plugging these values into the quadratic formula, we get:
L [5 ± (25 336)?]/2 [5 ± (361)?] / 2
This simplifies further to:
L [5 ± 19] / 2
This results in two possible solutions for the length:
L 24 / 2 12 meters
L -14 / 2 -7 meters
Since a negative length is not valid, we discard the negative solution. Therefore, the length of the bedroom is 12 meters.
Now, we can find the width:
Width L - 5 12 - 5 7 meters
Final Dimensions
The dimensions of the bedroom are:
Length: 12 meters Width: 7 metersAlternative Methods
Another approach to solving this problem involves using the quadratic formula directly without needing to set up the equation step-by-step. Here's a brief overview:
Given:
x x - 5 84
x2 - 5x - 84 0
(x - 12)(x 7) 0
x 12 or x -7 (since width can't be negative, we choose x 12)
Therefore:
Length 12 meters, Width 7 meters
Conclusion
Using quadratic equations can solve a wide range of practical problems in real-life situations. By understanding the underlying principles and applying them step-by-step, you can find the dimensions of a variety of shapes and structures.