The product of two consecutive integer is 5 three times the larger integer, what are the integers?
To solve this problem, let's break it down step by step:
Let's assume the smaller integer is x.
According to the problem statement, the larger integer will be the next consecutive integer, which is x + 1.
The product of two consecutive integers is 5 three times the larger integer, which can be written as:
x(x + 1) = 5 * 3(x + 1)
Now, let's solve this equation:
x(x + 1) = 15(x + 1)
Expanding both sides of the equation:
x^2 + x = 15x + 15
Rearranging the equation:
x^2 + x - 15x - 15 = 0
Combining like terms:
x^2 - 14x - 15 = 0
Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. Since the quadratic equation can't be easily factored, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -14, and c = -15:
x = (-(-14) ± √((-14)^2 - 4 * 1 * -15)) / (2 * 1)
Simplifying the equation:
x = (14 ± √(196 + 60)) / 2
x = (14 ± √256) / 2
x = (14 ± 16) / 2
Now, we have two possible values for x:
1. x = (14 + 16) / 2 = 30 / 2 = 15
2. x = (14 - 16) / 2 = -2 / 2 = -1
Therefore, the two consecutive integers are either 15 and 16 or -1 and 0.