Find the Range of $$\tfrac{x^2 + 3x + 5}{x^2 + x-6}$$

Question

anonymous · Answer

(x^2 +3x + 5) /  (x^2 + x - 6)

experimentx · Answer

try factorizing the denominator

experimentx · Answer

x^2 + x - 6 = (x+3)(x-2)
check for range in this interval
(-inf, -3),(-3,2),(2, inf)

anonymous · Answer

then...)

experimentx · Answer

what is the min value and max value of the expression interval (-inf, -3)?

anonymous · Answer

y = (x^2 +3x + 5) /  (x^2 + x - 6)


y(x^2 + x - 6)=(x^2 +3x + 5)

anonymous · Answer

x^2 y + xy -6y = x^2 + 3x + 5


x^2 (y-1) + x( y-3)  -  (6y + 5) = 0


(y-3)^2 + 4 (y-1) (6y+5) >=0

anonymous · Answer

is this correct.) @experimentX

anonymous · Answer

y^2 - 6y + 9 + 4 ( 6y^2 + 5y - 6y - 5) > =0

25y^2 - 10 y -11 > =0

anonymous · Answer

and i am stuck here......)

experimentx · Answer

x^2 (y-1) + x( y-3)  -  (6y + 5) = 0
x is always real ... 
so (y-3)^2 + 4(y-1)(6y+5) >= 0

anonymous · Answer

Yup....i have taken it...)

anonymous · Answer

25y^2 - 10 y -11 > =0 ............

i cant factorize this....)

experimentx · Answer

anonymous · Answer

ok....) thxxx.....

experimentx · Answer

seems you did your work correctly.

experimentx · Answer