find the values of m for which x2+3xy+x+my-m has two linear factors in x and y, with interger coefficients.
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x^2 +3xy +x +my -m =0
the standard form of this equation ---
x^2+0.y^2 +3xy +2.(1/2)x +2(m/2)y+(-m)=0
this is now just like
ax^2+by^2+2hxy+2gx+2fy+c =0
according to question this equation break into two linear equation.
hence,
we use ,
∆ =abc+2fgh-af^2-bg^2-ch^2 =0
=1.0.(-m)+2(m/2)(1/2)(3/2)-1(m/2)^2-0.(1/2)^2-(-m)(3/2)^2=0
0 +3m/4 -m^2/4 +9m/4 =0
12m -m^2 =0
m(m-12)=0
m=0,12
the standard form of this equation ---
x^2+0.y^2 +3xy +2.(1/2)x +2(m/2)y+(-m)=0
this is now just like
ax^2+by^2+2hxy+2gx+2fy+c =0
according to question this equation break into two linear equation.
hence,
we use ,
∆ =abc+2fgh-af^2-bg^2-ch^2 =0
=1.0.(-m)+2(m/2)(1/2)(3/2)-1(m/2)^2-0.(1/2)^2-(-m)(3/2)^2=0
0 +3m/4 -m^2/4 +9m/4 =0
12m -m^2 =0
m(m-12)=0
m=0,12
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