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Chapter 4: Quadratic Equations - Forming Equations from Word Problems

Introduction:
In this chapter, we will discuss how to form a quadratic equation from a word problem correctly. Quadratic equations are expressed as ax^2 + bx + c = 0, where a, b, and c are constants. These equations are commonly used to solve problems involving quadratic relationships or finding unknown values.

Step 1: Understand the Problem
Before forming the equation, it is crucial to thoroughly understand the given word problem. Identify the key information and variables involved, as well as any given conditions or relationships.

Step 2: Define the Variables
Assign variables to the unknown quantities mentioned in the problem. Typically, we use letters such as x, y, or z to represent these unknowns.

Step 3: Identify the Relationships
Determine the relationships between the given quantities. Pay attention to keywords or phrases that indicate mathematical operations such as addition, subtraction, multiplication, or division. Additionally, note any conditions or constraints mentioned in the problem.

Step 4: Translate the Information into Equations
Using the identified relationships, translate the given information into equations. Here are some common scenarios and their corresponding equations:

1. Sum or Difference of Two Quantities:
- If the problem states that the sum of two numbers is a specific value, the equation can be represented as x + y = c.
- Similarly, if the problem mentions that the difference between two numbers is a given value, the equation can be expressed as x - y = c.

2. Area or Perimeter:
- If the problem involves the area of a rectangle, which is given by the formula A = length × width, the equation becomes A = lw.
- For the perimeter of a rectangle, given by P = 2(length + width), the equation becomes P = 2(l + w).

3. Quadratic Relationships:
- If the problem describes a quadratic relationship, such as the area of a square or the height of an object thrown vertically, the equation will have the form ax^2 + bx + c = 0.

Step 5: Simplify and Solve
Once the equation is formed, simplify it by combining like terms, if necessary. To solve the equation, apply methods like factoring, completing the square, or using the quadratic formula.

Example:
Let's consider the following word problem:
"The sum of two consecutive integers is 45. Find the two integers."

Step 1: Understand the Problem
We are given that the sum of two consecutive integers is 45.

Step 2: Define the Variables
Let x be the first integer, and x + 1 be the consecutive integer.

Step 3: Identify the Relationships
The problem states that the sum of these two numbers is 45.

Step 4: Translate the Information into Equations
The sum of the two consecutive integers can be represented as x + (x + 1) = 45.

Step 5: Simplify and Solve
Simplifying the equation, we get 2x + 1 = 45. By subtracting 1 from both sides, we have 2x = 44. Dividing by 2, we find x = 22. Therefore, the two consecutive integers are 22 and 23.

Conclusion:
Forming equations from word problems involves understanding the problem, defining variables, identifying relationships, and translating the information into equations. By following these steps