SOLUTION: "The square of a positive number is six more than five times the positive number. Find the number." I'm confused because we did not do this type of problem as an example in class.

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Question 260018: "The square of a positive number is six more than five times the positive number. Find the number."
I'm confused because we did not do this type of problem as an example in class.
We are required to write a let statement. So i have:
Let n=the positive number
and then from my let statement and the word problem given, i took a guess and did this:
n(squared) = 5n + 6
Is this correct? And if so... how do i go on solving it?
Found 2 solutions by richwmiller, Earlsdon:
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
n^2=5n+6
Looks good
n^2-5n-6=0
find the factors
set the factors equal to 0
and solve the two factors for n
one will be positive and one will be negative you wan the positive x (The factor will have a minus in it)
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression n%5E2-5n-6, we can see that the first coefficient is 1, the second coefficient is -5, and the last term is -6.



Now multiply the first coefficient 1 by the last term -6 to get %281%29%28-6%29=-6.



Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient -5?



To find these two numbers, we need to list all of the factors of -6 (the previous product).



Factors of -6:

1,2,3,6

-1,-2,-3,-6



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -6.

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:



First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1




From the table, we can see that the two numbers 1 and -6 add to -5 (the middle coefficient).



So the two numbers 1 and -6 both multiply to -6 and add to -5



Now replace the middle term -5n with n-6n. Remember, 1 and -6 add to -5. So this shows us that n-6n=-5n.



n%5E2%2Bhighlight%28n-6n%29-6 Replace the second term -5n with n-6n.



%28n%5E2%2Bn%29%2B%28-6n-6%29 Group the terms into two pairs.



n%28n%2B1%29%2B%28-6n-6%29 Factor out the GCF n from the first group.



n%28n%2B1%29-6%28n%2B1%29 Factor out 6 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28n-6%29%28n%2B1%29 Combine like terms. Or factor out the common term n%2B1



===============================================================



Answer:



So n%5E2-5%2An-6 factors to %28n-6%29%28n%2B1%29.



In other words, n%5E2-5%2An-6=%28n-6%29%28n%2B1%29.



Note: you can check the answer by expanding %28n-6%29%28n%2B1%29 to get n%5E2-5%2An-6 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
What you have done so far is absolutely correct!
n%5E2+=+5n%2B6 You should be able to recognise this as a quadratic equation (the variable n is squared) so let's get into the standard form for quadratic equations.
n%5E2-5n-6+=+0 This factorable so...
%28n%2B1%29%28n-6%29+=+0 Apply the zero product rule.
n%2B1+=+0 or n-6+=+0 which means that...
n+=+-1 or n+=+6
Now since the problem specifies that "The square of a positive number is..."
you can discard the negative solution (n = -1) and your answer is n = 6.