How to factor 5n^2-56n+60

Question

I need help with this math problem please. And just help with factoring trinomials with a number infront of the first number

Answer 1

5n² - 56n + 60

 

Look at the coefficient of the square term... 5

and the constant ... 60

If we multiply those we have (5)(60) = 300

 

If we can find factors of 300 that add to the coefficient of the n term ... -56

Then we can factor by grouping:

Looking at the factors of 300 we have   300   1

                                                           60   5

                                                           50   6

 

We can stop right there because obviously we can get 56 from 50 and 6

The factors of 300 that add to -56 are (-50)(-6)

 

Replace the middle term -56n with -50n - 6n and rewrite the equation.

 

5n² - 56n + 60 = 5n² - 50n - 6n + 60

Grouping these we have  (5n²-50n) + (-6n + 60)

                                   = 5n(n - 10) -6(n-10)

The idea is to get the same thing in each set of parentheses as we have done...

5n(n-10)-6(n-10) = (5n-6)(n-10)                     

Answer 2

The same factoring concept is used even though you now have a value in front of the first variable.

 

With the additional number, you will utilize that as part of your factoring.

 

(nx +/- a) (nx +/- b) = 5n^2-56n+60

where n equals the factors of the number in front

 

The factors of 5 are:

 

     1      5

 

If you look at the factors of 60 you get:

 

      1     60

      2     30

      3     20

      4     15

      5     12

      6     10

 

Which values when multiplied together will give you a +60 but added together give you a -56?

 

Since the last value 60 is positive, the factors must either be both positive or both negative.

 

The middle value is a negative, which are added together.  Since adding two positive numbers cannot give you a negative value, the factors of 60 must be both negative.

 

Now which combination of factors will give you a total of 56 when multiplied out and added together?  You can eliminate some pretty quickly.

 

     1   5   and   1   60 will not be able to give us 56.  (n - 1) (5n - 60) would give us 65

     1   5   and   2   30 will not be able to give us 56.  (n - 2) (5n - 30) would give us 40

     1   5   and   3   20 will not be able to give us 56.  (n - 3) (5n - 20) would give us 35

     1   5   and   4   15 will not be able to give us 56.  (n - 4) (5n - 15) would give us 35

     1   5   and   5   12 will not be able to give us 56.  (n - 5) (5n - 12) would give us 37

     1   5   and   6   10 can give us 56.       

 

(n - 10) (5n - 6)

 

Multiplying it out using FOIL, you get:

 

5n^2 - 6n - 50n +60, and reducing it gives us:

 

5n^2 - 56n + 60 which matches the orignal equation.

 

Hope that helps!

Answer 3

Hi Ashley,

 

So we know that one of the factors needs to be 5n and the other is n since 5n times n = 5n^2.  

 

We also know that the second term of each factor has to be negative since the multiplier for n is -56 and they must both be negative since the unit term is +60.  

 

If it was -60 then one would be negative and the other positive. so our starting point is (5n-?1)(n-?2).

 

so the two have to multiply to 60 and one multiplied with 5n and the other with n sum to -56n.

 

Factors of 60 could be 1 and 60

or 2 and 30

or 3 and 20

or 5 and 12

or 6 and 10

 

So the one that jumps out is 5*(-10n) -6n = -56n

 

Thus the answer is (drumroll)  ==>   (5n-6)(n-10)

 

I hope this was helpful.

Sincerely,

Ralph Kettell