Problem 36 - A History of Mathematics: An Introduction

Question

Solve the following problem from BM \(13901:\) I added onethird of the square- side to two-thirds of the area of the square, and the result was \(0 ; 20(=1 / 3)\). Find the squareside.

Step-by-Step Solution

Answer

Answer: The length of the square side is 11/2.

Step 1: Define the variables

Let x be the length of the square side, and A be the area of the square.

Step 2: Write the equation

Using the given information, we can write the equation: \(\frac{1}{3}x + \frac{2}{3}A = \frac{61}{3}\).

Step 3: Express the area in terms of the square side

Since the area of a square is the square of its side, we have \(A = x^2\).

Step 4: Substitute A in the equation

Replace A with \(x^2\) in the equation: \(\frac{1}{3}x + \frac{2}{3}x^2 = \frac{61}{3}\).

Step 5: Multiply through by 3 to get rid of denominators

Multiply every term by 3: \(x + 2x^2 = 61\).

Step 6: Rearrange the equation into a quadratic equation

Rearrange the equation into a quadratic equation: \(2x^2 + x - 61 = 0\).

Step 7: Solve the quadratic equation

Now, we can solve the quadratic equation. We can factor the equation as: \((2x - 11)(x + 6) = 0\). Thus, we have two possibilities: \(x = \frac{11}{2}\) or \(x = -6\).

Step 8: Determine the valid solution

Since the length of a square side cannot be negative, we can eliminate \(x = -6\), leaving us with \(x = \frac{11}{2}\).

Step 9: Conclusion

The length of the square side is \(\frac{11}{2}\).

Key Concepts

Mathematical Problem Solving

Understanding mathematical problem solving is crucial to navigating through various math challenges, such as those posed by quadratic equations. In the context of the given problem, the first step was to carefully analyze the given information and identify what we are solving for - in this case, the length of a square's side. Logical reasoning and algebraic skills come into play to construct an equation, step by step, using the defined variables.

Once the variables were defined, translating the word problem into a mathematical equation became possible. This not only involves pure algebra but also an understanding of how arithmetic operations relate to the real-world scenario presented in the problem. In the step-by-step solution outlined, the process involved addition, understanding the concept of fractions, and a substitution method, which replaced the area of the square with a variable expression. Such techniques are fundamental to problem-solving in mathematics, where breaking down complex problems into simpler steps can lead us to a solution.

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, which means they include a variable raised to the power of two. They generally take the form of \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). The given problem required solving a real-world problem by formulating and solving a quadratic equation.

In the provided solution, manipulation of the initial equation resulted in a quadratic equation \(2x^2 + x - 61 = 0\). Following that, factoring is one of the methods employed to solve quadratic equations, which might lead to two possible solutions. However, considering the context, negative or unrealistic solutions should be discarded. Ultimately, knowing how to solve a quadratic equation is pivotal to finding the correct dimension of a square in such scenarios.

Area of a Square

The area of a square, an essential geometric concept, is calculated as the side length squared, expressed mathematically as \(A = x^2\). It represents the amount of space enclosed within the square. Relating area to the side length of the square is a foundational principle that was utilized in the provided problem to convert an area expression into a side-length variable expression.

By understanding the relationship between the area and side length of a square, the problem was transformed into an equation that correctly represents the given scenario. This transformation is a key strategy when tackling geometry-related questions within problem-solving exercises, enabling students to bridge the gap between geometric shapes and algebraic equations. Knowing how to connect geometric properties to algebraic expressions is a critical skill when dealing with mathematical problems involving shapes and dimensions.