Problem 148 - Elementary Algebra

Question

The product of two consecutive even numbers is 528 . Find the numbers.

Step-by-Step Solution

Answer

The numbers are 22 and 24.

Step 1: Identify the variables

Let the first even number be represented by the variable \( x \). The next consecutive even number is \( x + 2 \).

Step 2: Set up the equation

Since the product of the two consecutive even numbers is 528, write the equation as \( x(x + 2) = 528 \).

Step 3: Expand the equation

Expand the equation to form a quadratic equation: \( x^2 + 2x = 528 \).

Step 4: Move all terms to one side

Subtract 528 from both sides to set the equation to zero: \( x^2 + 2x - 528 = 0 \).

Step 5: Factor the quadratic equation

Find the factors of the quadratic equation. In this case, \( (x + 24)(x - 22) = 0 \).

Step 6: Solve for \( x \)

Set each factor equal to zero: \( x + 24 = 0 \) and \( x - 22 = 0 \). This gives the solutions \( x = -24 \) and \( x = 22 \).

Step 7: Determine the even numbers

Since even numbers must be positive for this context, select \( x = 22 \). The consecutive even number is \( x + 2 = 24 \).

Step 8: Verify the solution

Check the solution by multiplying the numbers: \( 22 \times 24 = 528 \), which is correct.

Key Concepts

quadratic equations

Quadratic equations are equations where the highest power of the variable is squared (i.e., raised to the power of 2). They generally have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are constants, and \( x \) is the variable.
In this exercise, our quadratic equation is formed by expanding the product of two consecutive even numbers: \( x(x + 2) = 528 \) leading to \( x^2 + 2x - 528 = 0 \). Every equation of this form represents a parabola when graphed, and can have up to two real solutions.
These solutions are called the roots of the equation, and they can be found using various methods such as factoring, completing the square, or the quadratic formula.

factoring

Factoring is a process used to break down a quadratic equation into simpler expressions that can be set to zero to find the solutions. In our exercise, we need to factor \( x^2 + 2x - 528 = 0 \).
We look for two numbers that multiply to give the constant term (-528) and add to give the coefficient of the middle term (2). These numbers in our case are 24 and -22.
Thus, the equation factors into \( (x + 24)(x - 22) = 0 \).
This factorization makes it easy to solve for the variable by setting each factor to zero, leading to our next step of solving the equation by finding the values of \( x \).

solving algebraic equations

Solving algebraic equations involves finding the value of the unknown variable(s) that make the equation true. For quadratic equations, once they've been factored, each factor is set to zero because if a product of two numbers is zero, at least one of them must be zero.
For example, from \( (x + 24)(x - 22) = 0 \), set each factor to zero, getting \( x + 24 = 0 \) and \( x - 22 = 0 \).
Solving these equations, we find \( x = -24 \) and \( x = 22 \). These are our potential solutions. It's essential to verify which solutions fit the context of the original problem, ensuring to meet conditions such as the numbers being positive even numbers.

variables

Variables are symbols used to represent unknown values in mathematical equations and expressions. In our exercise, the variable \( x \) represents the first of our two consecutive even numbers. By defining \( x \) and understanding the relationship between consecutive even numbers, we can model the problem accurately.
The exercise shows that the second even number is then expressed as \( x + 2 \).
It is important to consistently use variables throughout an exercise to avoid confusion and ensure each step logically follows from the last, making it easier to solve complex equations.