What is the smallest of 3 consecutive positive integers if the product of the smaller two integers is 5 less than 5 times the largest integer? I can't remember how to start this.
let the integers be a-1,a,a+1
(a-1)a = 5(a+1)-5
a = 6
the numbers are 5,6,7
check: 5*6 = 5*7-5
To solve this problem, let's break it down step-by-step:
Step 1: Define the variables:
Let's denote the three consecutive positive integers as x, x+1, and x+2.
Step 2: Set up the equation:
According to the problem statement, the product of the smaller two integers is 5 less than 5 times the largest integer. Mathematically, this can be written as:
x * (x+1) = 5(x+2) - 5
Step 3: Simplify the equation:
Expand the right side:
x^2 + x = 5x + 10 - 5
Combine like terms:
x^2 + x = 5x + 5
Step 4: Rearrange the equation:
Bring all terms to one side to form a quadratic equation:
x^2 + x - 5x - 5 = 0
Simplify:
x^2 - 4x - 5 = 0
Step 5: Solve the equation:
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
Factor the quadratic equation:
(x - 5)(x + 1) = 0
Setting each factor equal to zero, we have:
x - 5 = 0 or x + 1 = 0
Solve for x:
If x - 5 = 0, then x = 5
If x + 1 = 0, then x = -1
Since we're looking for positive integers, we discard the solution x = -1.
Therefore, the smallest of the three consecutive positive integers is x = 5.