Q1) α is the root of equation x2 = x – 2. Prove that α4 = 2 – 3α.
Q2) α is the root of equation x2 = 2x – 3. Prove α3 = α – 6 and α2 – 2 α3 = 9.
Q3) α is the root of equation 2x2 = 3x – 4. Prove 4 α 3 = α – 12.
1)
a^2= a-2
square both sides
a^4= a^2-4a+4
a^4= (a-2) - 4a +4
a^4 = 2 - 3a (proven)
2)
a^2 = 2a -3
multiply by a
a^3 = 2a^2 -3a
a^3 = 2(2a-3) -3a
a^3 = a -6 (Proven)
-2a^3 = -2(a-6)
a^2-2a^3 = a^2 -2a +12
a^2-2a^3 = (2a-3) -2a +12
a^2- 2a^3 = 9 (proven)
3)
2a^2= 3a -4
Multiply by 2a
4a^3= 6a^2 - 8a
4a^3 = 3(3a-4) - 8a
4a^3 = a -12 (proven)