Problem 145 - Elementary Algebra

Question

The product of two consecutive even numbers is 624 . Find the numbers.

Step-by-Step Solution

Answer

The numbers are 24 and 26.

Step 1: Identify the variables

Let the two consecutive even numbers be denoted as \(x\) and \(x + 2\).

Step 2: Formulate the equation

Since the product of the two numbers is 624, the equation can be written as: \(x(x + 2) = 624\)

Step 3: Expand the equation

Expand the equation to get a quadratic form: \(x^2 + 2x - 624 = 0\)

Step 4: Solve the quadratic equation using the quadratic formula

To solve the quadratic equation \(x^2 + 2x - 624 = 0\), use the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), where \(a = 1\), \(b = 2\), and \(c = -624\).

Step 5: Calculate the discriminant

Calculate the discriminant \(\Delta\) first: \(\Delta = b^2 - 4ac = 2^2 - 4(1)(-624) = 4 + 2496 = 2500\)

Step 6: Find the roots of the equation

Now, substitute \(\Delta\) back into the quadratic formula: \(x = \frac{{-2 \pm \sqrt{2500}}}{2(1)} = \frac{{-2 \pm 50}}{2}\). Hence, the roots are \(x = \frac{48}{2} = 24\) and \(x = \frac{-52}{2} = -26\).

Step 7: Determine the even numbers

Since \(x = 24\) is non-negative, the two consecutive even numbers are 24 and 26. (The negative root is not relevant for positive consecutive even numbers).

Key Concepts

Consecutive Even Numbers

Consecutive even numbers are even numbers that follow one another without any odd number in between. For example, 2, 4, 6, and 8 are consecutive even numbers.
When dealing with algebra problems involving consecutive even numbers, we denote the first even number as a variable, say \( x \). The next consecutive even number will then be \( x + 2 \).

Let's apply this to the given problem. Here, we denote the first even number as \( x \) and the second one as \( x + 2 \).
We are given that their product is 624, so we write this relationship as an equation:
\[ x(x + 2) = 624 \] This marks the beginning of solving the quadratic equation for these even numbers.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It states that the solutions for \( x \) can be found using:
\[ x = \frac{{-b \, \text{±} \, \sqrt{b^2 - 4ac}}}{2a} \] Here, \( a \), \( b \), and \( c \) are coefficients from the original quadratic equation. This formula can solve any quadratic equation, even if the roots are not real numbers.

To apply this to our problem, we first rewrite the equation \( x^2 + 2x - 624 = 0 \). Here, the coefficients are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -624 \)
Plug these values into the quadratic formula to find:\[ x = \frac{{-2 \, \text{±} \, \sqrt{2500}}}{2} \].
After calculating the equation, we obtain two roots which are \( x = 24 \) and \( x = -26 \).
Since we are looking for positive consecutive even numbers, we select 24. Thus, our even numbers are 24 and 26.

Discriminant Calculation

The discriminant of a quadratic equation helps determine the nature of the roots of the equation. The discriminant, denoted by \( \Delta \), is calculated using:
\[ \Delta = b^2 - 4ac \]
This calculation serves the following purposes:
- If \( \Delta > 0 \), there are two distinct real roots.
- If \( \Delta = 0 \), there is exactly one real root (or a repeated root).
- If \( \Delta < 0 \), the roots are complex and not real numbers.
In our problem, we have:
\[ b^2 = 2^2 = 4 \]
\[ 4ac = 4 \(1\)\(-624\) = -2496 \]
The discriminant calculation becomes:
\[ \Delta = 4 - (-2496) = 4 + 2496 = 2500 \]
Since \( \Delta = 2500 \gt 0 \), there are two distinct real roots.
Using these roots in the quadratic formula, we find the even numbers in question, ensuring that our solution is complete and accurate.