f.4 maths

Question

Let y=
2x+5
_____.Find the range of values of y for all real values of x.
x^2+6
Hence,find the range of values of I Y I .

Answer 1

y=
2x+5
_____
x^2+6
y(x^2+6)=2x+5
yx^2+6y=2x+5
yx^2-2x+(6y-5)=0
Δ greater than of equal to 0
(-2)^2-4y(6y-5) greater than of equal to 0
4-24y^2+20y greater than of equal to 0
-24y^2+20y+4 greater than of equal to 0
24y^2+20y+4 smaller than of equal to 0
∴ y between -1/3 and-1/2(可以等於-1/3 或-1/2)

∴| y | between 1/3 and 1/2(可以等於1/3 或 1/2)

Answer 2

y=
2x+5
_____
x^2+6
y(x^2+6)=2x+5
yx^2+6y=2x+5
yx^2-2x+(6y-5)=0
Δ greater than or equal to 0 ( b^2 - 4ac ) greater than or equal to 0

4 - (4)(y)(6y-5) greater than or equal to 0
4 - 24y^2 + 20y greater than or equal to 0
0 greater than or equal to 24y^2 - 20y - 4
0 greater than or equal to 6y^2 - 5y - 1
0 greater than or equal to (y + 1)(y - 6)


*************************** -1 ***************** 6
y+1 ************** - ***************** + *************** +
y-6 *************** - ***************** - **************** +
(y + 1)(y - 6) **** + **************** - **************** +

-1 smaller than or equal to y smaller than or equal to 6

0 smaller than or equal to |y| smaller than or equal to 6

2006-10-24 22:42:48 補充：
y between -1 and 6|y| between 0 and 6

2006-10-24 22:51:29 補充：
I make error in the calculation, it should be as follow0 greater than or equal to 6y^2 - 5y - 10 greater than or equal to (6y + 1)(y - 1)y between -(1/6) and 1|y| between 0 and 1