Word Problems with Quadratics – Physics, Business, and Geometry Applications

1. Introduction In this lesson, you will learn how to solve real-world word problems using quadratic equations. We will explore how descriptions of geometry, physics, and finance are modeled by second-degree polynomial equations and how to translate these problems into standard algebraic form.

2. Core Concept Explanation Word problems require translating English sentences into algebraic relationships. * Geometry: Area of rectangles (A = l \times w) often involves one side defined in terms of another (e.g. l = w + 4), creating a quadratic relationship: w(w + 4) = A. * Physics (Projectile Motion): The height of a falling object over time is modeled by h(t) = -gt^2 + v_0t + h_0, which is a quadratic equation. * Business (Revenue): Revenue is price times quantity sold. If price changes affect the quantity sold linearly, revenue becomes a quadratic function: R(x) = (p_0 - x)(q_0 + kx).

3. Step-by-Step Method To solve quadratic word problems: 1. Identify the unknown variable and define it clearly (e.g., "Let w be the width of the field"). 2. Translate the description into algebraic terms: * "Product" means multiply. * "Consecutive numbers" means x and x + 1. * "Consecutive odd or even numbers" means x and x + 2. 3. Set up the equation and write it in standard form: ax^2 + bx + c = 0. 4. Solve the equation using factorization, completing the square, or the quadratic formula. 5. Check constraints: in physical applications, lengths, times, and counts cannot be negative. Reject any negative roots that do not make physical sense.

4. Worked Examples * Example 1: The product of two consecutive positive odd integers is 63. Find the numbers. * Solution: 1. Let the first positive odd number be x. The next consecutive positive odd number is x + 2. 2. Write the product relation: x(x + 2) = 63 \Rightarrow x^2 + 2x - 63 = 0. 3. Solve: (x + 9)(x - 7) = 0 \Rightarrow x = -9 or x = 7. 4. Since the question specifies positive integers, reject -9. 5. The numbers are 7 and x+2 = 9. * Example 2: A rectangular lawn has a length 4 meters longer than its width. If the area of the lawn is 96 square meters, find its dimensions. * Solution: 1. Let the width be w meters. The length is w + 4 meters. 2. Area relation: w(w + 4) = 96 \Rightarrow w^2 + 4w - 96 = 0. 3. Solve by factorization: (w + 12)(w - 8) = 0 \Rightarrow w = -12 or w = 8. 4. Reject w = -12 since a width cannot be negative. 5. Dimensions are width = 8 meters and length = 12 meters. * Example 3: A ball is thrown upward with an initial velocity of 14 \text{ m/s}. Its height after t seconds is given by h = 14t - 5t^2. When does it hit the ground? * Solution: 1. The ball hits the ground when h = 0. 2. Set 14t - 5t^2 = 0 \Rightarrow t(14 - 5t) = 0. 3. Solve: t = 0 (at launch) or 14 - 5t = 0 \Rightarrow t = 2.8 seconds. 4. The ball hits the ground after 2.8 seconds.

5. Exam Relevance * Example WAEC Question: The sum of the squares of two consecutive positive integers is 85. Find the numbers. * Solution: Let the consecutive integers be x and x + 1. x^2 + (x+1)^2 = 85 \Rightarrow x^2 + x^2 + 2x + 1 = 85 \Rightarrow 2x^2 + 2x - 84 = 0 Divide by 2: x^2 + x - 42 = 0 \Rightarrow (x + 7)(x - 6) = 0. Reject -7 as the integer must be positive. The integers are 6 and 7.

6. Common Mistakes * Keeping physically invalid solutions: Writing down a negative width or time as a final answer. Always discard the invalid negative root. * Setting up consecutive odd/even numbers incorrectly: Writing them as x and x + 1 instead of x and x + 2.

7. Practice Questions 1. The hypotenuse of a right-angled triangle is 10 cm. If the base is 2 cm shorter than the height, find the length of the base. * Answer: Base is 6 cm (height is 8 cm, solving x^2 + (x-2)^2 = 100). 2. The area of a rectangular card is 120 square centimeters. The length is 7 cm more than the width. Find the width. * Answer: Width is 8 cm (solving w(w+7) = 120).

8. Summary / Cheat Sheet * Consecutive integers: x, x+1. * Consecutive odd/even integers: x, x+2. * Always write the final dimensions with units (e.g. meters, seconds). * Discard invalid negative solutions in geometry and physics contexts.