Question

How many integer values are there for k for which 4x2 + kxy – 9y2 is factorable?

Answer 1

To factor the expression we need to split the middle term into two parts.
Let k = p + q where k, p, q are all integers.
4x^2 + kxy - 9y^2 = 4x^2 + pxy + qxy - 9y^2 = (4x^2 + pxy) + (qxy - 9y^2) =
x(4x + py) + y(qx - 9y)
In order to factor we must have (4x + py) = (qx - 9y)
Or q = 4 and p = -9
Therefore, k = p + q = -9 + 4 = -5
That is one integer.

Answer 2

that is really confusing... can you maybe dumb the instructions down a bit?

Answer 3

I myself am not 100% sure if this is the way to proceed. But there is at least one integer found this way. Let me think if there may be other ways.

Answer 4

I heard the answer was 9 if that helps

Answer 5

wow!

Answer 6

what?

Answer 7

I still need help figuring out how to get 9

Answer 8

i got k = 0, ±35, ±16, ±9, ±5

Answer 9

What method did you use @souwing ?

Answer 10

the same way you did actually.

Given (4x^2 + pxy) + (qxy - 9y^2), and p+q = k
4x (x + (p/4)y) + qy (y - (9/q)y)

so to be factorable, p/4 = -9/q
pq = -36

so the solutions are the integer factors of -36. I.e

-1 and 36 or 1 and -36
-2 and 18 or 2 and -18
etc...

since k = q + p, just add up the factors

Answer 11

the k=0 case is easy — the middle terms drop out leaving you with a difference of squares which is of course factorable

Answer 12

easy in hindsight, that is :-)

Answer 13

In[57]:= eq = Out[49]

Out[57]= 4 x^2 + k x y - 9 y^2

In[58]:= Factor[eq /. k -> 0]

Out[58]= (2 x - 3 y) (2 x + 3 y)

In[59]:= Factor[eq /. k -> 35]

Out[59]= (4 x - y) (x + 9 y)

In[60]:= Factor[eq /. k -> -35]

Out[60]= (x - 9 y) (4 x + y)

In[62]:= Factor[eq /. k -> 16]

Out[62]= (2 x - y) (2 x + 9 y)

In[63]:= Factor[eq /. k -> -16]

Out[63]= (2 x - 9 y) (2 x + y)

In[64]:= Factor[eq /. k -> 9]

Out[64]= (4 x - 3 y) (x + 3 y)

In[65]:= Factor[eq /. k -> -9]

Out[65]= (x - 3 y) (4 x + 3 y)

In[66]:= Factor[eq /. k -> 5]

Out[66]= (x - y) (4 x + 9 y)

In[67]:= Factor[eq /. k -> -5]

Out[67]= (4 x - 9 y) (x + y)

Answer 14

if you had a need to see all of the factorings, you're all set :-)

Answer 15

sfaigan: If you need a simpler version without the p and q:

The original expression is: 4x2 + kxy – 9y2
If you multiply the coefficients of the first and last terms you get 4 * -9 = -36.
We need to split the middle term coefficient k such that its product is -36 much like the AC method of factoring a quadratic.

-36 can be factored as:
(-1)*36 or 1*(-36)
(-2)*18 or 2*(-18)
(-3)*12 or 3*(-12)
(-4)*9 or 4*(-9)
(-6)*6 or 6*(-6)

And the middle coefficient k will be the sum of the above factors.
k = 0, ±5, ±9, ±16, ±35

A total of 9 factors.