Concrete Slab Area Problem: Solving For X
Let's dive into a fascinating problem involving concrete slabs, areas, and a touch of algebra. This problem, categorized under mathematics, presents a scenario where we need to determine the dimensions of concrete slabs based on their combined area. We'll break down the problem step-by-step, making it easy to follow along and understand the underlying concepts. So, grab your thinking caps, and let's get started!
Problem Statement
The problem revolves around Aled, who possesses three concrete slabs. Two of these slabs are square in shape, each having a side length of x meters. The third slab is rectangular, with dimensions of 1 meter by (x + 1) meters. The combined area of these three slabs is given as 7 square meters. Our task is twofold:
(a) Demonstrate that the given information leads to the quadratic equation 2x² + x - 6 = 0.
(b) Solve the quadratic equation to find the value of x, which represents the side length of the square slabs.
This problem elegantly combines geometric concepts (areas of squares and rectangles) with algebraic techniques (forming and solving quadratic equations). By carefully analyzing the given information and applying relevant formulas, we can successfully navigate through the problem and arrive at the solution. Let's begin by dissecting the area calculations for each type of slab.
Part (a): Showing that 2x² + x - 6 = 0
The core of this part lies in expressing the total area of the slabs in terms of x and then equating it to the given total area, which is 7 . To achieve this, we need to calculate the individual areas of the square and rectangular slabs.
Area of the Square Slabs
Since there are two square slabs, and each has a side length of x meters, the area of one square slab is x * x = x² square meters. Therefore, the combined area of the two square slabs is 2 * x² = 2x² square meters. Remember, the area of a square is found by squaring the length of one of its sides. This fundamental concept is crucial for understanding the geometric aspect of the problem. The areas of geometric shapes form the basis for many mathematical and real-world applications, from calculating the space needed for construction materials to designing efficient layouts for rooms and buildings. Understanding how to calculate these areas accurately is a valuable skill.
Area of the Rectangular Slab
The rectangular slab has dimensions 1 meter by (x + 1) meters. The area of a rectangle is calculated by multiplying its length and width. Thus, the area of the rectangular slab is 1 * (x + 1) = (x + 1) square meters. This calculation is straightforward but essential for completing the total area equation. Just like with squares, the area of a rectangle is a fundamental geometric concept. It is used in countless practical applications, from determining the amount of fabric needed for a project to calculating the surface area of a field. Grasping the principles behind area calculations allows us to solve real-world problems with precision.
Total Area and the Equation
The total area of the three slabs is the sum of the areas of the two square slabs and the rectangular slab. This can be expressed as: Total Area = Area of two square slabs + Area of the rectangular slab. Substituting the expressions we derived earlier, we get: Total Area = 2x² + (x + 1) square meters. We are given that the total area is 7 square meters. Therefore, we can set up the equation: 2x² + x + 1 = 7. To arrive at the desired equation, we need to rearrange this equation by subtracting 7 from both sides: 2x² + x + 1 - 7 = 0. Simplifying this, we get: 2x² + x - 6 = 0. This is the quadratic equation we were asked to show. This step is a crucial bridge between the geometric representation of the problem and its algebraic formulation. By expressing the geometric relationships in terms of an equation, we can leverage the powerful tools of algebra to find a solution.
Part (b): Solving the Quadratic Equation 2x² + x - 6 = 0
Now that we have established the quadratic equation, we need to solve it for x. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward approach.
Factoring the Quadratic Equation
The quadratic equation is 2x² + x - 6 = 0. To factor this, we look for two numbers that multiply to give (2 * -6) = -12 and add up to 1 (the coefficient of the x term). These numbers are 4 and -3. We can rewrite the middle term (x) as 4x - 3x: 2x² + 4x - 3x - 6 = 0. Now, we factor by grouping: 2x(x + 2) - 3(x + 2) = 0. We can see that (x + 2) is a common factor, so we factor it out: (x + 2)(2x - 3) = 0. This factorization is a key step in solving the equation. It transforms the quadratic expression into a product of two linear factors, which makes it easier to find the values of x that satisfy the equation. Factoring requires recognizing patterns and applying algebraic manipulation skills. It's a fundamental technique in algebra that has wide-ranging applications.
Finding the Values of x
For the product of two factors to be zero, at least one of them must be zero. Therefore, we set each factor equal to zero and solve for x: 1. x + 2 = 0 => x = -2 2. 2x - 3 = 0 => 2x = 3 => x = 3/2 = 1.5 We have obtained two possible values for x: -2 and 1.5. However, since x represents the length of the side of a concrete slab, it cannot be negative. Lengths are physical quantities and are always non-negative. Therefore, we discard the solution x = -2. This step highlights the importance of considering the context of the problem when interpreting mathematical solutions. In real-world applications, not all solutions obtained through mathematical processes are necessarily valid. We must always assess whether the solutions make sense within the given context.
The Valid Solution
The only valid solution is x = 1.5 meters. This means that the square slabs each have sides of 1.5 meters, and the rectangular slab has dimensions 1 meter by (1.5 + 1) = 2.5 meters. We can verify this solution by plugging x = 1.5 back into the original equation for the total area: 2*(1.5)² + (1.5 + 1) = 2*2.25 + 2.5 = 4.5 + 2.5 = 7 square meters. This confirms that our solution is correct. This verification step is a valuable practice. It ensures that the solution we have obtained not only satisfies the equation but also aligns with the conditions of the original problem. Checking our work helps to prevent errors and build confidence in our results.
Conclusion
In this problem, we successfully demonstrated how geometric concepts and algebraic techniques can be combined to solve real-world problems. We started by calculating the areas of the concrete slabs, formed a quadratic equation based on the total area, and then solved the equation to find the side length of the square slabs. The process involved understanding geometric formulas, manipulating algebraic expressions, and interpreting the solutions in the context of the problem. This problem serves as a good example of how mathematical skills can be applied to practical situations. From construction and engineering to design and planning, mathematics plays a crucial role in solving a wide range of problems. By mastering the fundamental concepts and techniques, we can tackle complex challenges and make informed decisions.
For further exploration of quadratic equations and their applications, you can visit Khan Academy's Quadratic Equations section.
