Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 65347 by Masumsiddiqui399@gmail.com last updated on 28/Jul/19

Answered by MJS last updated on 28/Jul/19

$$\mathrm{squaring}(\Rightarrow \mathrm{beware}\mathrm{of}\mathrm{false}\mathrm{solutions}!)$$$$\Rightarrow$$$$(x^2+x-1)(x^3+x^2-2x-1)=0$$$$x^2+x-1=0\Rightarrow x=-\frac{1}{2}\pm \frac{\sqrt{5}}{2}$$$$x^3+x^2-2x-1=0$$$$x=z-\frac{1}{3}$$$$z^3-\frac{7}{3}z-\frac{7}{27}=0$$$$\Rightarrow$$$$z_1=-\frac{2\sqrt{7}}{3}\sin \left(\frac{1}{3}\arcsin \frac{\sqrt{7}}{14}\right)$$$$z_2=\frac{2\sqrt{7}}{3}\sin (\frac{\pi }{3}+\frac{1}{3}\arcsin \frac{\sqrt{7}}{14})$$$$z_3=-\frac{2\sqrt{7}}{3}\cos (\frac{\pi }{6}+\frac{1}{3}\arcsin \frac{\sqrt{7}}{14})$$$$\Rightarrow x_i=-\frac{1}{3}+z_i$$$$\mathrm{testing}\mathrm{all}\mathrm{these}\mathrm{with}\mathrm{the}\mathrm{original}\mathrm{equation}$$$$\Rightarrow$$$$x_1=-\frac{1}{2}-\frac{\sqrt{5}}{2}\approx -1.61803$$$$x_2=-\frac{1}{3}+\frac{2\sqrt{7}}{3}\sin (\frac{\pi }{3}+\frac{1}{3}\arcsin \frac{\sqrt{7}}{14})\approx 1.24698$$$$x_3=-\frac{1}{3}-\frac{2\sqrt{7}}{3}\cos (\frac{\pi }{6}+\frac{1}{3}\arcsin \frac{\sqrt{7}}{14})\approx -1.80194$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com