The product of 2 consecutive even whole numbers is 48. What is the larger whole number?
Answer (2)
Let x be the smaller even number, so the next consecutive even number is x + 2 .
Set up the equation x ( x + 2 ) = 48 .
Solve the quadratic equation x 2 + 2 x − 48 = 0 by factoring to get ( x + 8 ) ( x − 6 ) = 0 .
Choose the positive solution x = 6 , and find the larger number x + 2 = 8 . The final answer is 8 .
Explanation
Understanding the Problem We are given that the product of two consecutive even whole numbers is 48. Our goal is to find the larger of these two numbers.
Setting up the Equation Let x be the smaller even whole number. Then the next consecutive even whole number is x + 2 . The product of these two numbers is given by the equation: x ( x + 2 ) = 48
Expanding the Equation Expanding the equation, we get: x 2 + 2 x = 48
Rearranging to Quadratic Form Rearranging the equation into a standard quadratic form, we have: x 2 + 2 x − 48 = 0
Factoring the Quadratic We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -48 and add to 2. These numbers are 8 and -6. Thus, we can factor the quadratic as: ( x + 8 ) ( x − 6 ) = 0
Finding Possible Solutions This gives us two possible solutions for x :
x + 8 = 0 ⇒ x = − 8 x − 6 = 0 ⇒ x = 6
Choosing the Correct Solution Since we are given that the numbers are whole numbers, we discard the negative solution x = − 8 . Therefore, the smaller even whole number is x = 6 .
Finding the Larger Number The larger even whole number is x + 2 = 6 + 2 = 8 .
Final Answer Thus, the larger of the two consecutive even whole numbers is 8.
Examples
Understanding consecutive even numbers and their products can be useful in various scenarios, such as optimizing resource allocation or designing symmetrical patterns. For instance, if you are arranging items in rows and columns where the number of rows and columns are consecutive even numbers, knowing their product helps in planning the layout efficiently. This concept also applies in financial planning, where understanding growth patterns in even increments can aid in predicting future outcomes.
The larger of the two consecutive even whole numbers whose product is 48 is 8. The smaller even number is 6, and together they fit the criteria given in the problem. Therefore, 8 is the final answer.
;
