Solve algebraically using one variable: Find three consecutive odd integers such that the

product of the first integer and the third integer is equal to nine more than twelve times the middle integer.

(x-2)(x+2) = 9+12x

x = 13

So, the numbers are 11,13,15

Consecutive odd have a difference of two.

1st = n 2nd = n + 2 3rd =n + 4

n(n + 4) = 9 + 12(n+2)

n^2 + 4n = 9 + 12n + 24

n^2 -8n -33 = 0

(n -11)(n+3) = 0

n - 11 = 0 n + 3 = 0
n = 11 or n = -3

11, 13, 15 or -3,-1, 0

If you use these choices to do the check; 11, 13 and 15 work.

n (n+2) (n+4)

n(n+4) = 9 +12(n+2)

n^2 + 4 n = 9 + 12 n + 24

n^2 - 8 n - 33 = 0

(n-11)(n+3) = 0

n = 11
n+2 = 13
n+4 = 15

To solve this problem algebraically, let's represent the three consecutive odd integers:

Let the first odd integer be x.
The second odd integer will be x + 2 since it is consecutive.
The third odd integer will be x + 4 since it is consecutive as well.

According to the given information, the product of the first integer and the third integer is equal to nine more than twelve times the middle integer. Hence, the equation can be formed as:

x * (x + 4) = 12(x + 2) + 9

Now let's solve this equation step by step:

Step 1: Simplify both sides of the equation.

x^2 + 4x = 12x + 24 + 9

Step 2: Combine like terms on the right side.

x^2 + 4x = 12x + 33

Step 3: Move all terms to one side to set the equation to zero.

x^2 + 4x - 12x - 33 = 0

Step 4: Combine like terms on the left side.

x^2 - 8x - 33 = 0

Now we have a quadratic equation. To solve it, we can use factoring, completing the square, or the quadratic formula.

Step 5: Factor the quadratic equation.

(x - 11)(x + 3) = 0

Setting each factor to zero gives us two possible solutions:

x - 11 = 0 or x + 3 = 0

Solving for x in each case:

For x - 11 = 0:
x = 11

For x + 3 = 0:
x = -3

Step 6: Verify the solutions.

We need to check if the solutions satisfy the given conditions of being consecutive odd integers.

When x = 11:
The consecutive odd integers are 11, 13, and 15.
11 * 15 = 12(13) + 9
165 = 156 + 9 (matches)

When x = -3:
The consecutive odd integers are -3, -1, and 1.
-3 * 1 = 12(-1) + 9
-3 = -12 + 9 (doesn't match)

Therefore, the solution is x = 11, and the three consecutive odd integers are 11, 13, and 15.