SOLUTION: x^2 −y^2 + mx + 5y − 6 = (ax + by +c)(dx + ey + f). Determine the values for a, b, c, d, e, f, and m.

Algebra ->  Functions -> SOLUTION: x^2 −y^2 + mx + 5y − 6 = (ax + by +c)(dx + ey + f). Determine the values for a, b, c, d, e, f, and m.      Log On


   

Question 1127489: x^2 −y^2 + mx + 5y − 6 = (ax + by +c)(dx + ey + f). Determine the values for a, b, c, d, e, f, and m.
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2+-y%5E2+%2B+mx+%2B+5y+-6+=+%28ax+%2B+by+%2Bc%29%28dx+%2B+ey+%2B+f%29...first multiply right side of equation


compare coefficients and you see that:

ad=1->only when a=1 and d=1
be=-1->only when b=-1 and e=1
-m=af%2Bcd....since a=1 and d=1
-m=f%2Bc....eq.1
ae%2Bbd=0 since there is no xy term on left side
cf=-6 ...only if c=3 and f=-2 or c=1 and f=-6 or vice versa
I will go with c=3 and f=-2
then
-m=f%2Bc....eq.1
-m=-2%2B3
-m=1
m=-1
so, your solutions are:

a=1
b=-1
c=3
d=1
e=1
f=-2
m=-1


Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
She only gave one solution.  Also she made some illegal assumptions. There
are 8 altogether, although there are really only 4, because the two parentheses
on the right can be in reverse order.  That would be a, b and c trading places
with d, e and f, respectively. For instance, solutions 1 and 3 below are
essentially the same.  Here are all 8 possibilities (4 essentially): 

1.   x² - y² + x + 5y - 6 = (-x - y + 2)(-x + y - 3), in which
     a=-1, b=-1, c=2, d=-1, e=1, f=-3, m=1

2.   x² - y² - x + 5y - 6 = (-x - y + 3)(-x + y - 2), in which
     a=-1, b=-1, c=3, d=-1, e=1, f=-2, m=-1

3.   x² - y² + x + 5y - 6 = (-x + y - 3)(-x - y + 2), in which
     a=-1, b=1, c=-3, d=-1, e=-1, f=2, m=1

4.   x² - y² - x + 5y - 6 = (-x + y - 2)(-x - y + 3), in which
     a=-1, b=1, c=-2, d=-1, e=-1, f=3, m=-1

5.   x² - y² - x + 5y - 6 = (x - y + 2)(x + y - 3), in which
     a=1, b=-1, c=2, d=1, e=1, f=-3, m=-1

6.   x² - y² + x + 5y - 6 = (x - y + 3)(x + y - 2), in which
     a=1, b=-1, c=3, d=1, e=1, f=-2, m=1

7.   x² - y² - x + 5y - 6 = (x + y - 3)(x - y + 2), in which
     a=1, b=1, c=-3, d=1, e=-1, f=2, m=-1

8.   x² - y² + x + 5y - 6 = (x + y - 2)(x - y + 3), in which
     a=1, b=1, c=-2, d=1, e=-1, f=3, m=1

Edwin