Question

Find the separate equations form the lins represented by 3x^2 + 10xy + 8y^2 + 14x + 22y + 15

Answer 1

3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = 0

3x^2 is factorable only as 3x and x.
15 can be 15 * 1 or 3 * 5.
As a first guess, I place 5 and 3 in that order.
The coefficients of y are unknown so far.

= (3x + Ay + 5)(x + By + 3)

= 3x^2 + 3Bxy + 9x + Axy + ABy^2 + 3Ay + 5x + 5By + 15

(3B + A)xy = 10xy

(3A + 5B)y = 22y

  A + 3B = 10
3A + 5B = 22

-3A - 9B = -30
3A + 5B = 22

4B = 8

B = 2

A + 3B = 10

A + 6 = 10

A = 4


3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = (3x + 4y + 5)(x + 2y + 3)

The product of (3x + 4y + 5)(x + 2y + 3) is indeed
3x^2 + 10xy + 8y^2 + 14x + 22y + 15


The lines are
3x + 4y + 5 = 0
x + 2y + 3 = 0

Answer 2

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