SOLUTION: The expression 6y^2-y-51 can be rewritten as (3Ay+B)(y-C), where A, B, and C are positive integers. Find $ (AC)^2-B.

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Question 1083871: The expression 6y^2-y-51 can be rewritten as (3Ay+B)(y-C), where A, B, and C are positive integers. Find $
(AC)^2-B.
Found 3 solutions by Boreal, ikleyn, MathTherapy:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
6y^2-y-51 turns into y^2-y-306
Factor that into (y-18)(y+17), then divide the constant by 6 and reduce fully to (y-3)(y+17/6)
the factors then are (y-3) and (6y+17)
A=2
B=17
C=3
(AC)^2=36
36-17=19 is the answer.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
(3Ay + B)*(y - C) =  = .


Since it is identical to , we have


    3A = 6  and hence A = 2;             (1)

    B - 3AC = -1,   or  B - 6C = -1;     (2)

    BC = 51.                             (3)


Thus you actually have these two equations to determine B and C:

B - 6C = -1                              (2)
BC = 51.                                 (3)


From (2), express B = 6C -1 and substitute it into (3). You will get

(6C-1)*C = 51.


 = 0,

Factor left side

(C-3)*(2C+17) = 0.


Since you need C to be positive number (as the condition requires), you have only one possibility: C = 3.

Then B = 6C-1 = 6*3-1 = 17,

and now you have everything to calculate   =  =  = 19.



Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

The expression 6y^2-y-51 can be rewritten as (3Ay+B)(y-C), where A, B, and C are positive integers. Find $
(AC)^2-B.
 FACTORS to: (6y + 17)(y - 3).

Since  can be rewritten as: (3Ay + B)(y - C), we can then say that: (6y + 17)(y - 3) = (3Ay + B)(y - C)

By equating terms, we see that: 6y = 3Ay_____(6)y = (3A)y_____6 = 3A____

Also, B = 17, and - 3 = - C______3 = C

Thus,  

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