Questions: Find two consecutive odd integers such that their product is 63 more than 6 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 B. There are two solutions. The smaller consecutive odd integers are , and the larger consecutive odd integers are ?

Find two consecutive odd integers such that their product is 63 more than 6 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 B. There are two solutions. The smaller consecutive odd integers are , and the larger consecutive odd integers are ?

Transcript text: Find two consecutive odd integers such that their product is 63 more than 6 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 $\square$ B. There are two solutions. The smaller consecutive odd integers are $\square$ , and the larger consecutive odd integers are $\square$ ?

Solution

Solution Steps

To solve this problem, we need to find two consecutive odd integers, which we can represent as $ x $ and $ x+2 $. The problem states that their product is 63 more than 6 times their sum. We can set up an equation based on this condition and solve for $ x $.

Express the sum of the integers: $ x + (x + 2) = 2x + 2 $.
Express 6 times their sum: $ 6(2x + 2) = 12x + 12 $.
Set up the equation for the product: $ x(x + 2) = 12x + 12 + 63 $.
Simplify and solve the quadratic equation to find the value of $ x $.
Calculate the two consecutive odd integers using the value of $ x $.

Step 1: Define the Consecutive Odd Integers

Let the two consecutive odd integers be $ x $ and $ x+2 $.

Step 2: Set Up the Equation

According to the problem, the product of these integers is 63 more than 6 times their sum. Therefore, we can write the equation as: \[ x(x + 2) = 6(x + x + 2) + 63 \] Simplifying the right side, we have: \[ x(x + 2) = 12x + 12 + 63 \] \[ x(x + 2) = 12x + 75 \]

Step 3: Solve the Quadratic Equation

Rearrange the equation to form a standard quadratic equation: \[ x^2 + 2x = 12x + 75 \] \[ x^2 - 10x - 75 = 0 \] Solving this quadratic equation, we find the solutions: \[ x = -5 \quad \text{or} \quad x = 15 \]

Step 4: Determine the Consecutive Odd Integers

For $ x = -5 $, the consecutive odd integers are $-5$ and $-3$.

For $ x = 15 $, the consecutive odd integers are $15$ and $17$.