Question

The reals a and b are such that a3 - 3a2 + 5a = 1 and b3 – 3b2 + 5b = 5 are
satisfied. Find a + b.​

Answer 1

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Step-by-step explanation: Givena^3 – 3a^2 + 5a =1 and b^3 – 3b^2 + 5b =5

Soa^3 – 3a^2 + 3a-1=-2a

b^3 – 3b^2 + 3b-1 = -2b+4.

From 1,2 we have (a-1)3=-2a

(b-1)3=-2b+4....(4)

Let x=a-1 and y=b-1

So from 3 x3=-2(x+1) So x3+2x+2=0

From 4 we have y3=-2(y+1)+4

y3+2y-2=0....(6)

Adding 5 and 6 we get

x3+y3+2x+2y=0

x+y(x2-xy+y2)=0

So x+y=0 or x2-xy+y2=0

If x+y=0 we get a-1+b-1=0

So a+b=2