Question

solve the following equation by reducing them to form quadratic forms . 2^x+1 + 2^x +4^x = 4

Answer 1

${2}^{x + 1} + {2}^{x} + {4}^{x} = 4 \\ 2( {2}^{x} ) + {2}^{x} +( {2}^{x} ) {}^{2} = 4 \\ let \: {2}^{x} = t \\ eqn \: becomes \\ 2t + t + {t}^{2} = 4 \\ {t}^{2} + 3t - 4 = 0 \\ {t}^{2} + 4t - t - 4 = 0 \\ t(t + 4) - 1(t + 4) = 0 \\ (t - 1)(t + 4) = 0 \\ t = 1 \: and \: - 4 \\ we \: know \: \\ t = {2}^{x} \\ {2}^{x} = - 1 \: not \: possible \\ {2}^{x} = 4 \\ x = 2........ans$

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