Solving Consecutive Integer And Quadratic Equations Mathematical Problems

This article delves into the methods for solving two distinct mathematical problems. The first problem involves finding two consecutive integers whose squares sum up to 85. The second problem focuses on determining a number that, when multiplied by four less than twice itself, results in 96. We will explore the step-by-step solutions for each problem, providing a comprehensive understanding of the underlying algebraic principles.

Problem 1: Consecutive Integers

In this first problem, our main keyword centers around consecutive integers. We are presented with a classic algebraic puzzle: find two consecutive integers such that the sum of their squares equals 85. This problem provides an excellent opportunity to apply our knowledge of algebraic equations and problem-solving strategies. To effectively tackle this problem, we will employ a systematic approach, carefully defining our variables and setting up the equation. This methodical process not only leads us to the solution but also enhances our comprehension of mathematical relationships. Understanding this problem involves translating word problems into mathematical equations, solving quadratic equations, and applying the concept of consecutive integers.

Setting up the Equation

To solve this problem, we'll start by defining our variables. Let x represent the first integer. Since we're looking for consecutive integers, the next integer will be x + 1. Now, we need to translate the problem's statement into an algebraic equation. The problem states that the sum of the squares of these two integers is 85. This translates directly into the equation:

x² + (x + 1)² = 85

This equation forms the foundation for solving the problem. It encapsulates the relationship between the two consecutive integers and the given sum of their squares. The next step involves expanding and simplifying this equation to bring it into a more manageable form. This will allow us to identify the type of equation we're dealing with and select the appropriate solution method.

Solving the Quadratic Equation

Now, let's expand and simplify the equation: x² + (x + 1)² = 85. First, we expand the squared term: (x + 1)² = x² + 2x + 1. Substituting this back into our equation, we get:

x² + x² + 2x + 1 = 85

Combining like terms, we have:

2x² + 2x + 1 = 85

To solve this quadratic equation, we need to set it equal to zero. Subtracting 85 from both sides, we get:

2x² + 2x - 84 = 0

To simplify further, we can divide the entire equation by 2:

x² + x - 42 = 0

Now, we have a simplified quadratic equation in the standard form: ax² + bx + c = 0. We can solve this using factoring, the quadratic formula, or completing the square. In this case, factoring is the most straightforward approach.

Factoring the Quadratic

We need to find two numbers that multiply to -42 and add up to 1. These numbers are 7 and -6. Therefore, we can factor the quadratic equation as follows:

(x + 7)(x - 6) = 0

For the product of two factors to be zero, at least one of them must be zero. This gives us two possible solutions:

x + 7 = 0 or x - 6 = 0

Solving for x in each case, we get:

x = -7 or x = 6

These are the two possible values for the first integer. Remember, we're looking for consecutive integers, so we need to find the corresponding second integer for each value of x.

Finding the Consecutive Integers

We have two possible values for the first integer: x = -7 and x = 6. Let's find the corresponding consecutive integers for each value.

• Case 1: x = -7

  The next consecutive integer is x + 1 = -7 + 1 = -6. So, one pair of consecutive integers is -7 and -6.

• Case 2: x = 6

  The next consecutive integer is x + 1 = 6 + 1 = 7. So, another pair of consecutive integers is 6 and 7.

Therefore, the two pairs of consecutive integers whose squares sum to 85 are (-7, -6) and (6, 7). We can verify these solutions by squaring each integer and adding them together:

• (-7)² + (-6)² = 49 + 36 = 85
• (6)² + (7)² = 36 + 49 = 85

Both pairs satisfy the original problem statement.

Problem 2: The Product and Twice the Number

In this section, we tackle the second problem, which revolves around quadratic equations arising from a word problem. Our task is to find a number that, when multiplied by four less than twice that number, yields 96. This problem reinforces the importance of translating verbal descriptions into mathematical expressions. We will carefully break down the problem statement, identifying the key components and relationships, to construct the appropriate equation. The solution will involve solving a quadratic equation, which we'll approach using similar techniques as in the previous problem. This problem emphasizes the practical application of algebra in solving real-world scenarios, further solidifying our understanding of mathematical modeling.

Translating Words into an Equation

Our first step is to translate the word problem into a mathematical equation. Let's use y to represent the unknown number. The problem states that we need to multiply this number by