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 217381 by Rasheed.Sindhi last updated on 12/Mar/25

$$\mathrm{Let}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{be}\mathrm{real}\mathrm{numbers}\mathrm{satisfying}$$$$\mathrm{the}\mathrm{equations}$$$$\left(1\right)\mathrm{a}+\mathrm{b}+\mathrm{c}=4$$$$\left(2\right){\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=34$$$$\mathrm{Find}\mathrm{ab}+\mathrm{bc}+\mathrm{ca}$$

Commented by Tinku Tara last updated on 12/Mar/25

$$\mathrm{Two}\mathrm{equation}\mathrm{and}\mathrm{three}\mathrm{variables}$$$$\mathrm{there}\mathrm{are}\mathrm{infinitely}\mathrm{many}\mathrm{solutions}.$$$$$$

Commented by Tinku Tara last updated on 12/Mar/25

$$\mathrm{If}\mathrm{we}\mathrm{take}\mathrm{a}=0$$$$\mathrm{b}=2-\sqrt{\frac{3}{2}},\mathrm{c}=2+\sqrt{\frac{3}{2}}$$

Commented by Tinku Tara last updated on 12/Mar/25

$$ab+bc+ca=4-\frac{3}{2}=\frac{5}{2}$$$$\mathrm{The}\mathrm{solution}\mathrm{may}\mathrm{not}\mathrm{be}\mathrm{unique}.$$

Commented by Rasheed.Sindhi last updated on 13/Mar/25

$$\mathbb{T}\mathrm{han}\mathbb{k}\mathrm{s}\mathrm{Tinku}\mathrm{Tara}\mathrm{sir}!$$

Answered by Frix last updated on 12/Mar/25

$$\mathrm{Let}a=u-\sqrt{v}\wedge b=u+\sqrt{v}$$$$a,b\in \mathbb{R}\Rightarrow v⩾0$$$$\mathrm{We}\mathrm{get}$$$$v=u^2-8u+16-\frac{5}{u}⩾0\Rightarrow$$$$\frac{3-\sqrt{5}}{2}⩽u⩽\frac{3+\sqrt{5}}{2}\vee u⩾5$$$$ab+bc+ca=-4u^2+16u-16+\frac{5}{u}$$

Commented by Rasheed.Sindhi last updated on 13/Mar/25

$$\mathbb{N}\mathbf{i}\mathbb{C}\mathbf{e}!$$$$\mathcal{T}hankssir!$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com