Find the Value of x⁷ + 1/x⁷ From the Quadratic Equation
Quadratic Equation: If x² – x + 1 = 0, then find the value of x⁷ + 1/x⁷
Solution
we know, x² – x + 1 = 0
We can solve this quadratic equation
\begin{aligned} x&=\frac{1\pm \sqrt{1^2-4 \times 1 \times 1}}{2 \times 1} \\ \\ &=\frac{1\pm \sqrt{-3}}{2 } \\ \\ \Rightarrow \ x&=\frac{1\pm i\sqrt{3}}{2 } \\ \\ \end{aligned}
We got, x = (1 ± i√3)/2
If x = (1 – i√3)/2 then 1/x =(1 + i√3)/2
That is x and 1/x are roots of this quadratic equation
So, x⁷ + 1/x⁷ is a constant value
To solve this math problem we need to find x² first
x² = ((1 + i√3) / 2)² = (1 – 3 + 2i√3) / 4
x² = ( -2 + 2i√3)/4 = (-1 + i√3)/2 = –1/x
Now we get, x² = –1/x
⇒ x³ = -1
That is,
x⁷ + 1/x⁷ = (x³ × x³ × x) + 1/(x³ × x³ × x) = x + 1/x
⇒ x⁷ + 1/x⁷ = (1 – i√3)/2 + (1 + i√3)/2
⇒ x⁷ + 1/x⁷ = 1