The product of the smallest and largest of three consecutive even integers is three times the middle integer. Find the integers
x(x+4) = 3(x+2)
x^2 + 4x = 3x + 6
x^2 + x - 6 = 0
(x+3)(x-2) = 0
Let's represent the three consecutive even integers as x, x+2, and x+4.
The product of the smallest and largest of the three integers is (x)(x+4), and this is equal to three times the middle integer, which is 3(x+2).
So, we have the equation:
(x)(x+4) = 3(x+2)
Simplifying the equation:
x^2 + 4x = 3x + 6
Rearranging the equation:
x^2 + 4x - 3x - 6 = 0
x^2 + x - 6 = 0
Factoring the quadratic equation:
(x + 3)(x - 2) = 0
Setting each factor equal to zero:
x + 3 = 0 or x - 2 = 0
Solving for x:
x = -3 or x = 2
Since we are looking for consecutive even integers, we can eliminate the negative value.
Therefore, the three consecutive even integers are 2, 4, and 6.
To solve this problem, let's break it down step by step:
Step 1: Define the variables
Let's denote the three consecutive even integers as n, n + 2, and n + 4. Here, n represents the smallest even integer.
Step 2: Express the given information as equations
According to the problem, the product of the smallest and largest integers is three times the middle integer. Mathematically, this can be expressed as:
n * (n + 4) = 3 * (n + 2)
Step 3: Solve the equation
Expand the equation:
n^2 + 4n = 3n + 6
Rearrange the equation:
n^2 + n - 6 = 0
Factorize or use the quadratic formula to solve for n. Since this equation can be easily factored:
(n + 3)(n - 2) = 0
Set each factor equal to zero and solve:
n + 3 = 0 or n - 2 = 0
If n + 3 = 0, then n = -3
If n - 2 = 0, then n = 2
Step 4: Find the consecutive even integers
Since the problem specifies three consecutive even integers, we need to check if both solutions yield consecutive even numbers.
If n = -3, the consecutive even integers would be -4, -2, and 0.
If n = 2, the consecutive even integers would be 2, 4, and 6.
So, the two sets of integers that satisfy the problem are:
-4, -2, 0
2, 4, 6