Question Number 146914 by mathdanisur last updated on 16/Jul/21
$$\mid{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}\mid\:=\:\mid{x}−\mathrm{4}\mid\:\Rightarrow\:{x}=? \\ $$
Answered by liberty last updated on 16/Jul/21
$$\:\mid{a}\mid=\mid{b}\mid\:\Leftrightarrow\:\left({a}+{b}\right)\left({a}−{b}\right)=\mathrm{0} \\ $$$$\Rightarrow\left({x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}+{x}−\mathrm{4}\right)\left({x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}−{x}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow\left({x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{8}\right)\left({x}^{\mathrm{2}} −\mathrm{4}{x}\right)=\mathrm{0} \\ $$$$\Rightarrow\left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right){x}\left({x}−\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}\left({x}+\mathrm{2}\right)\left({x}−\mathrm{4}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow{x}=−\mathrm{2}\:;\:{x}=\mathrm{0}\:;\:{x}=\mathrm{4} \\ $$
Commented by mathdanisur last updated on 16/Jul/21
$${thank}\:{you}\:{Ser} \\ $$
Answered by Rasheed.Sindhi last updated on 17/Jul/21
$$\mid{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}\mid\:=\:\mid{x}−\mathrm{4}\mid\:\Rightarrow\:{x}=? \\ $$$$\begin{cases}{{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}=+\left({x}−\mathrm{4}\right)}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{OR}}\\{{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{4}=−\left({x}−\mathrm{4}\right)}\end{cases}\:\:\:\:\: \\ $$$$\begin{cases}{{x}^{\mathrm{2}} −\mathrm{4}{x}=\mathrm{0}\Rightarrow{x}\left({x}−\mathrm{4}\right)=\mathrm{0}}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{OR}}\\{{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{8}=\mathrm{0}\Rightarrow\left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right)=\mathrm{0}}\end{cases}\:\:\:\:\: \\ $$$${x}=\mathrm{0},\:−\mathrm{2},\:\mathrm{4} \\ $$
Commented by mathdanisur last updated on 18/Jul/21
$${Thank}\:{you}\:{Sir} \\ $$