Question Number 26680 by Sr@2004 last updated on 28/Dec/17
$$\left({b}−{c}\right){x}^{\mathrm{2}} +\left({c}−{a}\right){x}+\left({a}−{b}\right)=\mathrm{0}\:{if}\:{the}\: \\ $$$${eqation}\:{roots}\:\:{are}\:{eqal}\:{you}\:{proved}\:{that} \\ $$$$\mathrm{2}{b}={a}+{c}. \\ $$
Answered by mrW1 last updated on 28/Dec/17
$${equal}\:{roots}\:\Rightarrow\:\Delta=\mathrm{0} \\ $$$$\Delta=\left({c}−{a}\right)^{\mathrm{2}} −\mathrm{4}\left({b}−{c}\right)\left({a}−{b}\right)=\mathrm{0} \\ $$$$\Rightarrow{c}^{\mathrm{2}} −\mathrm{2}{ac}+{a}^{\mathrm{2}} −\mathrm{4}\left({ab}−{ac}−{b}^{\mathrm{2}} +{bc}\right)=\mathrm{0} \\ $$$$\Rightarrow{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −\mathrm{4}{ab}+\mathrm{2}{ac}+\mathrm{4}{b}^{\mathrm{2}} −\mathrm{4}{bc}=\mathrm{0} \\ $$$$\Rightarrow\left({a}+{c}\right)^{\mathrm{2}} −\mathrm{4}\left({a}+{c}\right){b}+\mathrm{4}{b}^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\left({a}+{c}−\mathrm{2}{b}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\mathrm{2}{b}={a}+{c} \\ $$