# Find the Value of x⁷ + 1/x⁷ From the Quadratic Equation

Quadratic Equation: If *x*² – *x* + 1 = 0, then find the value of *x*⁷ + 1/*x*⁷

## Solution

we know, *x*² – *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, *x*⁷ + 1/*x*⁷ is a constant value

To solve this math problem we need to find *x*² first

*x*² = ((1 + i√3) / 2)² = (1 – 3 + 2*i*√3) / 4

*x*² = ( -2 + 2*i*√3)/4 = (-1 + *i*√3)/2 = –1/*x*

Now we get, *x*² = –1/*x*

⇒ *x*³ = -1

That is,

*x*⁷ + 1/*x*⁷ = (*x*³ × *x*³ × *x*) + 1/(*x*³ × *x*³ × *x*) = *x* + 1/*x*

⇒ *x*⁷ + 1/*x*⁷ = (1 –* i*√3)/2 + (1 +* i*√3)/2

⇒ *x*⁷ + 1/*x*⁷ = 1