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Question Number 196652 by SANOGO last updated on 28/Aug/23

$$solve$$$$3x^{\frac{3}{2}}-4x^{\frac{1}{2}}+1=0$$

Answered by BaliramKumar last updated on 28/Aug/23

$$$$$$3{\left(\sqrt{x}\right)}^3-3\sqrt{x}-\sqrt{x}+1=0$$$$3\sqrt{x}[{\left(\sqrt{x}\right)}^2-1]-(\sqrt{x}-1)=0$$$$3\sqrt{x}\left[(\sqrt{x}-1)(\sqrt{x}+1)\right]-(\sqrt{x}-1)=0$$$$(\sqrt{x}-1)[3{\left(\sqrt{x}\right)}^2+3\sqrt{x}-1]=0$$$$\sqrt{x}-1=0or3{\left(\sqrt{x}\right)}^2+3\sqrt{x}-1=0$$$$\sqrt{x}=1\sqrt{x}=\frac{-3\pm \sqrt{21}}{6}$$$$x=1\sqrt{x}=\frac{-3+\sqrt{21}}{6}or\sqrt{x}=\frac{-3-\sqrt{21}}{6}$$$$\sqrt{x}=\sqrt{\frac{{(-3+\sqrt{21})}^2}{36}}\mathrm{invalid}$$$$\sqrt{x}=\sqrt{\frac{(30-6\sqrt{21})}{36}}$$$$\sqrt{x}=\sqrt{\frac{5-\sqrt{21}}{6}}$$$$x=\frac{5-\sqrt{21}}{6}$$

Answered by Frix last updated on 28/Aug/23

$$3x^{\frac{3}{2}}-4x^{\frac{1}{2}}+1=0$$$$x⩾0$$$$\mathrm{Obviously}x=1★$$$$\Rightarrow$$$$(\sqrt{x}-1)(x-\sqrt{x}-\frac{1}{3})=0$$$$\sqrt{x}=\frac{1}{2}+\frac{\sqrt{21}}{6}★$$

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