x1 and x2 are solutions of x^2-x+1=0. What is the value of the sum (x1^2-x1)^1000 + (x2^2-x2)^1000 ?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

x1 and x2 are given as the solutions of x^2 - x + 1 = 0

So if we substitute x with x1 we get x1^2 - x1 + 1 = 0

=> x1^2 - x1 = -1

Similarly with x2, x2^2 - x2 + 1 = 0

=> x2^2 - x2 = -1

The required sum is (x1^2-x1)^1000 + (x2^2-x2)^1000

=> -1^1000 + -1^1000

=> 1 + 1

=> 2

The required sum is 2.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We know the fact that the solution of an equation, substituted into equation, it cancels out the equation.

We'll substitute the solutions of the equation into the given equation:

x1^2 - x1 + 1 = 0

x1^2 - x1 = -1We'll raise to the power of 1000 both sides:

(x1^2-x1)^1000 = (-1)^1000

(x1^2-x1)^1000 = 1 (1)

x2^2 - x2 + 1 = 0

x2^2 - x2 = -1We'll raise to the power of 1000 both sides:

(x2^2-x2)^1000 = (-1)^1000

(x2^2-x2)^1000 = 1 (2)

We'll add (1) and (2):

(x1^2-x1)^1000 + (x2^2-x2)^1000 = 1 + 1

(x1^2-x1)^1000 + (x2^2-x2)^1000 = 2

The value of the sum = 2.

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