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.
To solve this problem, we need to set up an equation based on the given information and then solve for the smallest of the three consecutive positive integers.
Let's denote the three consecutive positive integers as x, x+1, and x+2 (where x is the smallest).
According to the problem statement, the product of the smaller two integers is 5 less than 5 times the largest integer. So, we can write the equation as:
x*(x+1) = 5*(x+2) - 5
Now, let's simplify and solve the equation step by step:
x^2 + x = 5x + 10 - 5
x^2 + x = 5x + 5
Rearranging the terms:
x^2 + x - 5x - 5 = 0
Combining like terms:
x^2 - 4x - 5 = 0
Now, we have a quadratic equation. To solve it, you can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula in this case:
x = [-b ± √(b^2 - 4ac)] / (2a)
For our equation x^2 - 4x - 5 = 0, a = 1, b = -4, and c = -5.
Plugging in the values, we get:
x = [-(-4) ± √((-4)^2 - 4(1)(-5))] / (2 * 1)
Simplifying further:
x = [4 ± √(16 + 20)] / 2
x = [4 ± √36] / 2
x = (4 ± 6) / 2
Now, we have two possible solutions:
x1 = (4 + 6) / 2 = 10 / 2 = 5
x2 = (4 - 6) / 2 = -2 / 2 = -1
Since we're looking for positive integers, the solution x = -1 is not valid. Therefore, the smallest of the three consecutive positive integers is x = 5.
So, the smallest integer is 5.