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Question Number 215092 by BaliramKumar last updated on 28/Dec/24

Commented by BaliramKumar last updated on 28/Dec/24

$$Iwanttofindthevalueof\mathit{a}and\mathit{b}$$

Commented by BaliramKumar last updated on 28/Dec/24

$$a^3-b^3=125$$$$a^2-b^2=25$$$$a+b=5$$$$a^2+ab+b^2=?$$

Answered by Rasheed.Sindhi last updated on 28/Dec/24

$$a^2-b^2=25\Rightarrow (a-b)(a+b)=25$$$$(a-b)\left(5\right)=25\Rightarrow a-b=5$$$$\{\begin{array}{l}a+b=5\\ a-b=5\end{array}\Rightarrow a=5,b=0$$$$a^2+ab+b^2=5^2+5\times 0+0^2=25$$

Commented by A5T last updated on 28/Dec/24

$$Wecanalsodothis,sowegetacontradiction.$$$$a^3-b^3=(a-b)(a^2+ab+b^2)=625$$$$a^2-b^2=(a-b)(a+b)=25$$$$a+b=5\Rightarrow a-b=\frac{25}{5}=5$$$$\Rightarrow a^2+ab+b^2=\frac{625}{5}=125$$

Commented by Frix last updated on 28/Dec/24

$${\mathrm{We}}^{\prime }\mathrm{ve}\mathrm{got}3\mathrm{equations}\mathrm{for}2\mathrm{unknowns}\mathrm{and}$$$$\mathrm{all}must\mathrm{be}\mathrm{satisfied}.$$$$\left(1\right)b=5-a$$$$\left(2\right)a^2=25+b^2$$$$a^2=25+{(5-a)}^2$$$$a^2=50-10a+a^2\Rightarrow a=5\Rightarrow b=0$$$$\left(3\right)\mathrm{Test}{a}^3-b^3=625\Rightarrow \mathrm{wrong}$$$$\Rightarrow$$$$\mathrm{No}\mathrm{solution}$$

Commented by BaliramKumar last updated on 29/Dec/24

$$ithinkquestioniswrong$$

Commented by Rasheed.Sindhi last updated on 29/Dec/24

$$Ithinkthatthequestionhascome$$$$beforethistimealso.$$

Commented by A5T last updated on 29/Dec/24

$$Q211393$$

Commented by Frix last updated on 29/Dec/24

$$\mathrm{A}\mathrm{question}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{wrong}.$$$$\mathrm{The}\mathrm{proposed}\mathrm{answers}\mathrm{are}\mathrm{wrong}.$$

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