Solve-for-x-R-x-3-3x-2-2-x-1-

Question Number 209872 by Frix last updated on 24/Jul/24

Solve for x \in R :

x^{3} - 3 x^{2} + 2 = \sqrt{x + 1}

Answered by Ghisom last updated on 24/Jul/24

x = t + 1

t^{3} - 3 t = \sqrt{t + 2}

expecting 3 solutions at most

remember \cos 3 α = {4 \cos}^{3} α - 3 \cos α

\Rightarrow t = 2 \cos α \land 0 ⩽ α ⩽ π

{8 \cos}^{3} α - 6 \cos α = \sqrt{2 + 2 \cos α}

2 \cos 3 α = \sqrt{2 (1 + \cos α)}

remember 1 + \cos 2 β = {2 \cos}^{2} β

\Rightarrow

2 \cos 3 α = \sqrt{{4 \cos}^{2} \frac{α}{2}}

\cos 3 α =∣ \cos \frac{α}{2} ∣

\Rightarrow α = 0 \lor α = \frac{4 π}{7} \lor α = \frac{4 π}{5}

\Rightarrow t = 2 \lor t = 2 \cos \frac{4 π}{7} \lor t = - \frac{1 + \sqrt{5}}{2}

\Rightarrow x = 3 \lor x = 1 - 2 \cos \frac{4 π}{7} \lor x = \frac{1 - \sqrt{5}}{2}

Commented by Frix last updated on 24/Jul/24

Great idea!

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