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Question Number 131298 by liberty last updated on 03/Feb/21

$$\mathrm{Let}{\cos }^{-1}\left(\mathrm{x}\right)+{\cos }^{-1}\left(2\mathrm{x}\right)+{\cos }^{-1}\left(3\mathrm{x}\right)=\pi$$$$.\mathrm{If}\mathrm{x}\mathrm{satisfies}\mathrm{the}\mathrm{cubic}{\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}-1=0$$$$\mathrm{then}\mathrm{a}+\mathrm{b}+\mathrm{c}\mathrm{has}\mathrm{the}\mathrm{value}\mathrm{equal}\mathrm{to}$$

Answered by mr W last updated on 03/Feb/21

$$\cos ({\cos }^{-1}x+{\cos }^{-1}2x)=\cos (\pi -{\cos }^{-1}3x)$$$$2x^2-\sqrt{(1-x^2)(1-4x^2)}=-3x$$$$\sqrt{(1-x^2)(1-4x^2)}=2x^2+3x$$$$(1-x^2)(1-4x^2)=4x^4+12x^3+9x^2$$$$12x^3+14x^2-1=0\equiv {ax}^3+{bx}^2+cx-1=0$$$$\Rightarrow a=12,b=14,c=0$$$$\Rightarrow a+b+c=26$$

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