Question Number 194190 by tri26112004 last updated on 29/Jun/23
$${Explanation}\:{Why}: \\ $$$${While}\:{f}\left({ax}+{b}\right)+{f}\left({cx}+{d}\right)={ex}+{g} \\ $$$${then}\:{f}\left({x}\right)={Ax}^{\mathrm{2}} +{Bx}+{C}\:¿ \\ $$
Commented by Tinku Tara last updated on 30/Jun/23
$$\mathrm{For}\:\mathrm{the}\:\mathrm{given}\:\mathrm{condition}\:\mathrm{it}\:\mathrm{is}\:\mathrm{not} \\ $$$$\mathrm{necessary}\:\mathrm{for}\:{f}\left({x}\right)\:\mathrm{to}\:\mathrm{be}\:\mathrm{quadratic}. \\ $$$${f}\left({x}\right)={Ax}+{B} \\ $$$${f}\left({ax}+{b}\right)={Aax}+{Ab}+{B} \\ $$$${f}\left({cx}+{d}\right)={Acx}+{Ad}+{B} \\ $$$${g}={A}\left({b}+{d}\right)+\mathrm{2}{B} \\ $$$${e}={A}\left({a}+{c}\right) \\ $$$${A}=\frac{{e}}{{a}+{c}} \\ $$$${B}={g}−\frac{{e}\left({b}+{d}\right)}{\left({a}+{c}\right)} \\ $$