Find two consecutive odd integers such that their product is 11 more than 2 times their sum.
Select the correct choice below and fill in the answer box(es) to complete your choice. (Use a comma to separate answers as needed.)
A. There is only one solution. The consecutive odd integers are ____.
Answer (2)
Define the consecutive odd integers as x and x + 2 .
Set up the equation: x ( x + 2 ) = 2 ( x + ( x + 2 )) + 11 .
Simplify the equation to a quadratic: x 2 − 2 x − 15 = 0 .
Solve the quadratic equation by factoring: ( x − 5 ) ( x + 3 ) = 0 , yielding x = 5 and x = − 3 . The two pairs of consecutive odd integers are 5 , 7 , − 3 , − 1 .
Explanation
Problem Analysis Let's analyze the problem. We are looking for two consecutive odd integers such that their product is 11 more than 2 times their sum. Let's denote the two consecutive odd integers as x and x + 2 .
Setting up the Equation Now, let's set up the equation according to the problem statement: The product of the two integers is x ( x + 2 ) , and their sum is x + ( x + 2 ) = 2 x + 2 . The problem states that the product is 11 more than 2 times their sum. So, we can write the equation as: x ( x + 2 ) = 2 ( 2 x + 2 ) + 11
Simplifying the Equation Next, we expand and simplify the equation: x 2 + 2 x = 4 x + 4 + 11 x 2 + 2 x = 4 x + 15 x 2 − 2 x − 15 = 0
Solving the Quadratic Equation Now, we solve the quadratic equation x 2 − 2 x − 15 = 0 . We can factor this equation as: ( x − 5 ) ( x + 3 ) = 0 So, the possible values for x are x = 5 and x = − 3 .
Checking the Solution (5 and 7) If x = 5 , the two consecutive odd integers are 5 and 7. Let's check if this solution satisfies the original condition: Product: 5 × 7 = 35 Sum: 5 + 7 = 12 2 × Sum + 11 = 2 × 12 + 11 = 24 + 11 = 35 Since the product equals 11 more than 2 times the sum, this solution is valid.
Checking the Solution (-3 and -1) If x = − 3 , the two consecutive odd integers are -3 and -1. Let's check if this solution satisfies the original condition: Product: ( − 3 ) × ( − 1 ) = 3 Sum: ( − 3 ) + ( − 1 ) = − 4 2 × Sum + 11 = 2 × ( − 4 ) + 11 = − 8 + 11 = 3 Since the product equals 11 more than 2 times the sum, this solution is also valid.
Final Answer Therefore, there are two solutions: the consecutive odd integers are 5 and 7, or -3 and -1.
Examples
Understanding consecutive odd integers and their relationships is useful in various mathematical puzzles and real-world scenarios. For instance, consider a problem where you need to arrange items in a grid with specific constraints on the number of items in adjacent rows or columns. The concept of consecutive odd integers can help you determine the possible arrangements that satisfy these constraints. Additionally, in financial modeling, understanding number patterns can assist in predicting trends or identifying anomalies in data sets.
The two consecutive odd integers that satisfy the condition are 5 and 7, and also -3 and -1. This is established by solving the equation formed from the problem statement. Therefore, the solutions are two pairs of integers: (5, 7) and (-3, -1).
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