Solve the following algebraically using one variable: Find three consecutive even integers such
that the product of the smallest integer and the middle integer is eight less than eleven times the larger integer.
even integers differ by 2, so the three numbers are x, x+2 and x+4. Now you are told that
x(x+2) = 11(x+4)-8
Now solve for x.
let n be the smallest integer
n+2 the middle
n+4 the larger
n*(n+2)+8=11(n+4)
solve for n. I appears to be a quadratic equation
n^2-9n-36=0
(n-12)(n+3)=0
n=12, or n=-3
reject the n=-3 as it is not an even integer.
To solve this problem algebraically, let's first represent the three consecutive even integers using variables.
Let's assume the smallest even integer is x. Since they are consecutive even integers, the middle integer would be x + 2 (since it is two numbers away from x) and the largest integer would be x + 4 (since it is four numbers away from x).
Now, let's set up the equation based on the given information.
The product of the smallest integer (x) and the middle integer (x + 2) is eight less than eleven times the larger integer (x + 4). So, the equation becomes:
x(x + 2) = 11(x + 4) - 8
Let's simplify the equation step by step.
x(x + 2) = 11x + 44 - 8
x^2 + 2x = 11x + 36
Rearranging the equation:
x^2 + 2x - 11x - 36 = 0
Combining like terms:
x^2 - 9x - 36 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Since the quadratic equation does not factor nicely, let's use the quadratic formula to find the solutions.
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = -9, and c = -36.
Substituting these values into the quadratic formula:
x = (-(-9) ± √((-9)^2 - 4(1)(-36))) / (2(1))
Simplifying further:
x = (9 ± √(81 + 144)) / 2
x = (9 ± √225) / 2
x = (9 ± 15) / 2
Now, we have two possible solutions for x:
x1 = (9 + 15) / 2 = 24 / 2 = 12
x2 = (9 - 15) / 2 = -6 / 2 = -3
Since we are looking for even integers, we can check if these solutions satisfy the condition.
For x = 12:
The three consecutive even integers would be 12, 14, 16. Let's check if the condition holds true:
12(14) = 11(16) - 8
168 = 176 - 8
168 = 168 (True)
For x = -3:
The three consecutive even integers would be -3, -1, 1. Let's check if the condition holds true:
-3(-1) = 11(1) - 8
3 = 11 - 8
3 = 3 (True)
Therefore, the solution to the problem is x = 12, and the three consecutive even integers are 12, 14, and 16.