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Question Number 131605 by physicstutes last updated on 06/Feb/21

$$\mathrm{solve}\mathrm{the}\mathrm{equation}$$$$m^4-7m^3+14m^2-7m+1=0$$

Answered by malwan last updated on 06/Feb/21

$$m^4+(-3m^3-4m^3)$$$$+(m^2+12m^2+m^2)$$$$+(-4m-3m)+1=0$$$$\Rightarrow (m^4-3m^2+m^2)$$$$+(-4m^3+12m^2-4m)$$$$+(m^2-3m+1)=0$$$$\Rightarrow m^2(m^2-3m+1)$$$$-4m(m^2-3m+1)$$$$+(m^2-3m+1)=0$$$$\Rightarrow (m^2-3m+1)(m^2-4m+1)=0$$$$m^2-3m+1=0$$$$\Rightarrow m=\frac{3\pm \sqrt{9-4}}{2}=\frac{3\pm \sqrt{5}}{2}$$$$or{m}^2-4m+1=0$$$$\Rightarrow m=\frac{4\pm \sqrt{16-4}}{2}=2\pm \sqrt{3}$$

Commented by physicstutes last updated on 06/Feb/21

$$\mathrm{thats}\mathrm{awsome}\mathrm{sir}$$

Commented by malwan last updated on 06/Feb/21

$$thanks$$

Answered by EDWIN88 last updated on 06/Feb/21

$$\left(1\right)({\mathrm{m}}^2+\mathrm{km}+1)({\mathrm{m}}^2+\ell \mathrm{m}+1)={\mathrm{m}}^4-{7\mathrm{m}}^3+{14\mathrm{m}}^2-7\mathrm{m}+1$$$$\Rightarrow {\mathrm{m}}^4+(\ell +\mathrm{k}){\mathrm{m}}^3+(2+\mathrm{k}\ell ){\mathrm{m}}^2+(\mathrm{k}+\ell )\mathrm{m}+1=$$$${\mathrm{m}}^4-{7\mathrm{m}}^3+{14\mathrm{m}}^2-7\mathrm{m}+1$$$$\Rightarrow \{\begin{array}{l}\mathrm{k}+\ell =-7\Rightarrow \mathrm{k}=-7-\ell \\ 2+\mathrm{k}\ell =14;\mathrm{k}\ell =12\end{array}$$$$\Rightarrow \ell (-7-\ell )=12;{\ell }^2+7\ell +12=0\to \{\begin{array}{l}\ell =-3\\ \mathrm{k}=-4\end{array}$$$$\mathrm{then}\iff ({\mathrm{m}}^2-3\mathrm{m}+1)({\mathrm{m}}^2-4\mathrm{m}+1)=0$$$$\iff [{(\mathrm{m}-\frac{3}{2})}^2-\frac{5}{4}][{(\mathrm{m}-2)}^2-3]=0$$$$\iff \mathrm{m}=\{\begin{array}{l}\frac{3}{2}\pm \frac{\sqrt{5}}{2}\\ 2\pm \sqrt{3}\end{array}.$$

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