Problem 25 - Elementary Algebra

Question

The product of two consecutive even integers is 168 . Find them.

Step-by-Step Solution

Answer

Question: Find two sets of consecutive even integers whose product is 168. Answer: The two sets of consecutive even integers are -14 and -12, or 12 and 14.

Step 1: Represent the integers as variables

Let x be the first even integer. Since consecutive even integers are two numbers apart, we can represent the second even integer as (x + 2).

Step 2: Set up the equation

The product of the two consecutive integers is 168, which means that: x(x + 2) = 168

Step 3: Expand the equation and solve for x

Expanding the equation we get: x^2 + 2x = 168 Move 168 to the left side and we have a quadratic equation: x^2 + 2x - 168 = 0

Step 4: Factor the quadratic equation

We should now factor this quadratic equation to find the possible values for x. This factoring can be done as follows: (x + 14)(x - 12) = 0

Step 5: Solve for x

Now we have two equations to find x: 1. x + 14 = 0 x = -14 2. x - 12 = 0 x = 12

Step 6: Find the consecutive even integers

After finding possible values for x, we can now find the consecutive even integers. If x = -14, the consecutive even integers are: First integer: -14 Second integer: -12 If x = 12, the consecutive even integers are: First integer: 12 Second integer: 14

Step 7: Verify the product

We should now check that the products of both sets of integers equal 168: (-14) × (-12) = 168 12 × 14 = 168 Both sets of consecutive even integers have a product of 168. Thus, our answer is correct, and the consecutive even integers are -14 and -12, or 12 and 14.

Key Concepts

Consecutive Integers

Understanding consecutive integers is simple. Consecutive integers are numbers that follow each other without any gaps. When dealing with even numbers, like in this exercise, each number is always separated by a difference of two.
In our problem, we are asked about consecutive even integers, where the first integer is represented by \( x \), and the next by \( x + 2 \). This way of setup is key in forming equations that can be solved.
By knowing the simple rule of consecutive numbers, especially even or odd ones, we make complex problems easier. This concept helps for algebraic expressions where you need to work with sequences of numbers.

Factoring

Factoring is a method used to solve quadratic equations. It involves expressing the equation as a product of two simpler expressions set to zero. In our problem, the equation \( x^2 + 2x - 168 = 0 \) is factored to find solutions.
To factor, look for two numbers that multiply to the constant term (-168) and add up to the linear coefficient (2). Here, the numbers 14 and -12 satisfy these conditions.
The factored form \( (x + 14)(x - 12) = 0 \) allows us to quickly find the values of \( x \) by setting each factor equal to zero. Factoring simplifies algebraic equations and is a powerful tool in solving quadratic problems.

Algebraic Equations

An algebraic equation involves a combination of numbers and variables that equals a specific value. In our case, \( x(x + 2) = 168 \) represents the product of two consecutive even integers.
Such equations can often be expanded or manipulated to find the values that satisfy them. By expanding, we get \( x^2 + 2x = 168 \), and rearranging gives a standard quadratic form \( x^2 + 2x - 168 = 0 \).
Algebraic equations form the backbone of solving real-world problems by using symbolic representation. They allow abstract thinking and precise calculations to determine unknown quantities. Practicing how to set up and solve these is fundamental in developing algebra skills.

Integer Solutions

When solving equations like the one in the exercise, we aim for integer solutions. These are whole numbers, such as -14 and 12, which are critical in real-life applications where fractional or decimal answers are not suitable.
The last step in such exercises involves checking if our solutions make sense within the context of the problem. In this case, both pairs of integers, -14 & -12 and 12 & 14, give a product of 168, confirming the correctness of our factors.
Understanding integer solutions is essential as they offer clarity and definite answers. This is often necessary in problems where elements like quantity, count, or age are involved, ensuring answers are logical and applicable.