SOLUTION: Is there a value of n such that n^2 + n + 1 is a multiple of 5?

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Question 711687: Is there a value of n such that n^2 + n + 1 is a multiple of 5?
Answer by Stitch(470)   (Show Source): You can put this solution on YOUR website!

Yes, there are values of n that produce multiples of 5. On the graph lines are drawn when Y=5, Y=10, Y=15 & Y=20. However the values of n that produce a multiple of 5 may not be whole numbers.
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For example we can solve for what value of n makes the equation equal 10.

Subtract 10 from both sides

Now we can use the quadratic equation to solve for n
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=37 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 2.54138126514911, -3.54138126514911. Here's your graph:

The quadratic shows us that when n is equal to -3.54 or 2.54, the equation is equal to 10, which is a multiple of five.
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