Solving area and dimensions problems with quadratic factoring.

📚 Understanding Area and Dimensions Problems with Quadratic Factoring

Many real-world problems involve finding the dimensions of a shape given its area. Often, these problems lead to quadratic equations, which can be solved by factoring. This guide will walk you through the process, providing a solid foundation for tackling these types of questions.

📜 Historical Context

The use of quadratic equations to solve geometric problems dates back to ancient civilizations, including the Babylonians and Greeks. They used geometric methods to solve what we now represent algebraically. Factoring, as a systematic method for solving quadratic equations, became more refined during the Islamic Golden Age and later in Europe.

🔑 Key Principles

• 📐 Formulating the Equation: Translate the word problem into a mathematical equation. Pay close attention to how the dimensions relate to each other (e.g., length is twice the width) and how they combine to give the area.
• ✍️ Setting up the Quadratic: Rearrange the equation into the standard quadratic form: $ax^2 + bx + c = 0$.
• ✂️ Factoring: Factor the quadratic expression. Look for two numbers that multiply to give $c$ and add to give $b$.
• 💡 Solving for x: Set each factor equal to zero and solve for $x$. These are the possible solutions.
• 🤔 Checking for Validity: Discard any solutions that don't make sense in the context of the problem (e.g., negative lengths).

✍️ Example 1: Rectangular Garden

A rectangular garden has an area of 75 square feet. The length of the garden is 2 feet more than twice the width. Find the dimensions of the garden.

1. 📝 Define variables: Let $w$ be the width and $l$ be the length.
2. 📐 Write equations: $l = 2w + 2$ and $A = lw = 75$.
3. ✍️ Substitute: $75 = (2w + 2)w$
4. ✂️ Expand and rearrange: $2w^2 + 2w - 75 = 0$
5. 💡 Factor: $(2w + 15)(w - 5) = 0$
6. ➗ Solve: $w = 5$ or $w = -\frac{15}{2}$.
7. 🤔 Check: Since width cannot be negative, $w = 5$. Therefore, $l = 2(5) + 2 = 12$.

The width of the garden is 5 feet, and the length is 12 feet.

🌳 Example 2: Increasing Dimensions

A square piece of cardboard is to be made into an open box by cutting 3-inch squares from each corner and folding up the sides. The box is to have a volume of 48 cubic inches. Find the original dimensions of the square.

1. 📝 Define variables: Let $x$ be the side length of the original square.
2. 📐 Write equations: The dimensions of the box are $(x - 6)$ by $(x - 6)$ by $3$. The volume $V = 3(x - 6)(x - 6) = 48$.
3. ✍️ Simplify: $(x - 6)^2 = 16$
4. ✂️ Expand and rearrange: $x^2 - 12x + 36 = 16 \implies x^2 - 12x + 20 = 0$
5. 💡 Factor: $(x - 10)(x - 2) = 0$
6. ➗ Solve: $x = 10$ or $x = 2$.
7. 🤔 Check: Since cutting 3-inch squares from a side of length 2 is impossible, $x = 10$.

The original dimensions of the square were 10 inches by 10 inches.

👷 Real-World Applications

• 🏠 Construction: Calculating the dimensions of rooms or buildings.
• 🏞️ Landscaping: Designing gardens or patios.
• 📦 Packaging: Optimizing the size of boxes and containers.
• 🎨 Art and Design: Creating scaled models or artwork.

🎯 Conclusion

Solving area and dimension problems using quadratic factoring is a valuable skill in mathematics and has many practical applications. By understanding the key principles and practicing with examples, you can master this technique and apply it to a wide range of problems.