Question Number 131525 by mathlove last updated on 05/Feb/21
$$\:{if}\:\:\:\left({a}−{b}\right)\left({a}+{b}\right)=\mathrm{23} \\ $$$${then}\:\:{faind}\:\:\:{a}\centerdot{b}=? \\ $$
Answered by mr W last updated on 05/Feb/21
$${no}\:{unique}\:{solution}\:{for}\:{a},{b}\in{R}. \\ $$$${for}\:{a},{b}\in{Z}: \\ $$$$\left({a}−{b}\right)\left({a}+{b}\right)=\mathrm{23}=\mathrm{1}×\mathrm{23} \\ $$$$\Rightarrow{a}−{b}=\mathrm{1} \\ $$$$\Rightarrow{a}+{b}=\mathrm{23} \\ $$$$\Rightarrow{a}=\frac{\mathrm{1}+\mathrm{23}}{\mathrm{2}}=\mathrm{12} \\ $$$$\Rightarrow{b}=\frac{\mathrm{23}−\mathrm{1}}{\mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow{a}×{b}=\mathrm{12}×\mathrm{11}=\mathrm{131} \\ $$$${or} \\ $$$${a}−{b}=−\mathrm{1} \\ $$$${a}+{b}=−\mathrm{23} \\ $$$$\Rightarrow{a}=−\mathrm{12} \\ $$$$\Rightarrow{b}=−\mathrm{11} \\ $$$$\Rightarrow{a}×{b}=\mathrm{131} \\ $$
Commented by mathlove last updated on 05/Feb/21
$${thanks} \\ $$
Commented by Rasheed.Sindhi last updated on 06/Feb/21
$${Also}\:{possible}: \\ $$$${a}+{b}=\mathrm{1}\:\wedge\:{a}−{b}=\mathrm{23} \\ $$$$\:\:\:\:\:{a}=\mathrm{12},{b}=−\mathrm{11} \\ $$$$\:\:\:\:\:{a}.{b}=−\mathrm{132} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{OR}} \\ $$$${a}+{b}=−\mathrm{1}\:\wedge\:{a}−{b}=−\mathrm{23} \\ $$$$\:\:\:\:\:{a}=−\mathrm{12},{b}=\mathrm{11} \\ $$$$\:\:\:\:\:{a}.{b}=−\mathrm{132} \\ $$
Commented by mr W last updated on 06/Feb/21
$${correct}\:{sir}!\:{thanks}\:{for}\:{remark}! \\ $$