Question
If x^2-x-1 = 0, then find the value of x^6
If x^2-x-1 = 0, then find the value of x^6
Using the root formula for a quadratic equation,
the roots for the equation
can be determined as
which are the Golden Ratios. The equation (1) is often referred as Golden Equation.
To obtain the value of x to the power of 6, we can do the following manipulation on Golden Equation.
First, moving -x-1 to the right hand side yields the expression represent x to the power of 2
Addition 2 to both sides of the Golden Equation
And then, multiply x+1 on both sides.
Now we can use difference of cubes to raise x to the power of 3
Then
Squaring both sides gives the expression to represent x to the power of 6
Now substitute x^2 in (3), and the value of x in (2) to (4)
So the final result of the value of x^6 is 9\pm4\sqrt{5}