A-Math問題

f(x)/g(x) = y = (x^2 + 2x - 2)/(-2x^2 - 12x - 23)
(-2x^2 - 12x - 23)y = x^2 + 2x - 2
-2yx^2 -12xy - 23y = x^2 + 2x - 2
(1+ 2y)x^2 + (2 + 12y)x + 23y - 2 = 0
For x to be real, delta > 0, that is
(2+12y)^2 - 4(1+2y)(23y -2) > 0
(1 + 6y)^2 - (1+2y)(23y - 2) > 0
1 + 36y^2 + 12y - 23y + 2 - 46y^2 + 4y > 0
-10y^2 - 7y + 3 > 0
10y^2 + 7y - 3 < 0
(10y - 3)(y +1) < 0
so -1< y < 3/10.
So the greatest value of y [=f(x)/g(x)] is 3/10 and the least value is -1.