Expert: Abe Mantell - 6/9/2009

Question
(Quadratic Equation)
1)Solve these equation.(use 2^x=y) 2^2x-9(2^x)+20=0

2)if p and q are the root for quadratic equation 2x²-5x-1=0,then find the value for
a)p²+q²
b)p over q + q over p

(Function)
3)The function f and g are defined by f:x=x²+3 and g:x=2x+1.Find the range of fg.  

4)The function f and g are defined by f:x=3x-1 and g:x=x²+1.Find the range of g.   

5)A function is defined by f(x)=x²-6,x>6. Find the valus of f(x) and f^-1(x) on the same axes.

Answer
Wow, you sure have many questions!  How about I do a couple and see if you can do the others?

1. Do you mean 2^(2x)-9(2^x)+20=0, if so then if y=2^x, we get
.  y^2-9y+20=0 ==> (y-4)(y-5)=0 ==> y=4,5 ==> 2^x=4 or 2^x=5
.  ==> x=2 or x=ln(5)/ln(2)

2. Divide 2x²-5x-1=0 by 2 to get x²-(5/2)x-1/2=0...if p & q solve this,
.  then it factors to (x-p)(x-q)=x^2-(p+q)x+pq...so, p+q=5/2 and pq=-1/2
.  If p+q=5/2, then (p+q)^2=25/4 ==> p^2+2pq+q^2=25/4...but since pq=-1/2
.  2pq=-1...so p^2-1+q^2=25/4 ==> p^2+q^2=29/4
.  p/q + q/p = (p^2+q^2)/(pq) = (29/4)/(-1/2) = -29/2

OK?

Abe