f4.....maths

a)g(1/2)=0
(1/2)b+1/2=0
(1/2)b=-1/2
b=-1
f(1/2)=0,subs b=-1 into f(1/2)=0
(1/2)^2+a(1/2-1)+1=0
1/4-(1/2)a+1=0
5/4-0.5a=0
0.5a=5/4
a=5/2

b)f(x)=0
x^2+a(x-1)-b=0-----(1)
subs a=5/2 b=-1 into (1)
x^2+5/2(x-1)+1=0
x^2+5x/2-3/2=0
x=1/2(reject) or-3
x=-3

c)f(x)=g(x)
x^2+5x/x-3/2=-x+1/2
x^2+7x/2-2=0
x=0.5or-4


a)f(x)=x^2-4x+1
f(a)=a^2+4a+1
f(b)=b^2+4b+1
f(a)=f(b)=0
a^2-4a+1=b^2+4b+1=0

b)a^2-4a+1=b^2+4b+1=0
a^2-4a+1-b^2-4b-1=0
a^2-b^2-4a+4b=0
(a-b)(a+b)-4(a-b)=0
(a-b)[a+b-4(1)]=0
(a-b)(a+b-4)=0
a-b=0 or a+b-4=0
a=b(rej.) or a+b=4

c)f(a+b)=(a+b)^2-4(a+b)+1
=4^2-4(4)+1
=1

(1)
A) f(1/2)=(1/2)^2+a(1/2-1)-b=0..........(1)
g(1/2)=b(1/2)+1/2 =0................(2)

by(1) 1/4-a/2-b=0.........(3)
by(2) b=-1.........(4)

(4)in(3) 1/4-a/2+1=0
a=5/2

B) f(x)=0
x^2+(5/2)(x-1)+1=0
x^2+(5/2)x-5/2+1=0
2x^2+5x-3=0
(2x-1)(x+3)=0
x=1/2 OR x=-3
SO the other root of f(x) =0 is x=-3

C) f(x)=x^2+(5/2)(x-1)+1
g(x)=-x+1/2
f(x)=g(x) ==> x^2+(5/2)(x-1)+1=-x+1/2
x^2+(5/2)x-5/2+1+x-1/2=0
x^2+(7/2)x-2=0
2x^2+7x-4=0
(2x-1)(x+4)=0
x=1/2 OR x=-4


(2)

A) f(a) =f(b) =0
so f(a)=a^2 - 4a +1=0
f(b)=b^2 - 4b +1=0

B) a^2 - 4a +1=0
a=2+2√3 OR 2-2√3

b^2 - 4b +1=0
b=2+2√3 OR 2-2√3

a≠b SO a+b=4.............ANS

C) f(a+b)=f(4)=4^2-4*4+1=1.............ANS