Solve-the-equation-in-R-2-x-2-x-1-1-x-x-

Question Number 155153 by mathdanisur last updated on 26/Sep/21

Solve the equation in R

\sqrt{2 (x^{2} - x + 1)} = 1 + \sqrt{x} - x

Answered by MJS_new last updated on 26/Sep/21

\sqrt{x} \in R \Rightarrow x ⩾ 0 but {it}^{'} s easy to see x \neq 0

and lhs > 0 \Rightarrow 1 + \sqrt{x} - x > 0

\Rightarrow

0 < x < \frac{3}{2} + \frac{\sqrt{5}}{2}

let x = t^{2} \land t > 0

\sqrt{2 (t^{4} - t^{2} + 1)} = - t^{2} + t + 1 [\Rightarrow 0 < t < \frac{1}{2} + \frac{\sqrt{5}}{2}]

squaring

2 (t^{4} - t^{2} + 1) = {(t^{2} - t - 1)}^{2}

t^{4} + 2 t^{3} - t^{2} - 2 t + 1 = 0

{(t^{2} + t + 1)}^{2} = 0

\Rightarrow t = - \frac{1}{2} + \frac{\sqrt{5}}{2}

\Rightarrow x = \frac{3}{2} - \frac{\sqrt{5}}{2}

Commented by mathdanisur last updated on 26/Sep/21

awesome solution, thank you S er