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arcsin-x-2-3-arcsin-x-2-3x-4-x-




Question Number 207272 by hardmath last updated on 10/May/24
arcsin (x^2  − 3) = arcsin (x^2  + 3x + 4)  x = ?
$$\mathrm{arcsin}\:\left(\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3}\right)\:=\:\mathrm{arcsin}\:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{3x}\:+\:\mathrm{4}\right) \\ $$$$\boldsymbol{\mathrm{x}}\:=\:? \\ $$
Commented by mr W last updated on 10/May/24
no solution! since x^2 +3x+4≥1.75>1.
$${no}\:{solution}!\:{since}\:{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{4}\geqslant\mathrm{1}.\mathrm{75}>\mathrm{1}. \\ $$
Commented by Frix last updated on 11/May/24
But the equation is true for x=−(7/4) with  lhs=rhs=arcsin ((22)/9) =(π/2)−i ln ((22+(√(403)))/9)
$$\mathrm{But}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{is}\:\mathrm{true}\:\mathrm{for}\:{x}=−\frac{\mathrm{7}}{\mathrm{4}}\:\mathrm{with} \\ $$$$\mathrm{lhs}=\mathrm{rhs}=\mathrm{arcsin}\:\frac{\mathrm{22}}{\mathrm{9}}\:=\frac{\pi}{\mathrm{2}}−\mathrm{i}\:\mathrm{ln}\:\frac{\mathrm{22}+\sqrt{\mathrm{403}}}{\mathrm{9}} \\ $$

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