Question

Resolva a equação
$5 {}^{x} + 125.5 {}^{ - x} = 30$

Answer 1

$5^x+125.5^{-x}=30\\ \\ 5^x+{125\over5^x}=30$

$5^x=y\\ \\ y+{125\over y}=30\\ \\ mmc=y\\ \\ y^2+125=30y\\ \\ y^2-30y+125=0$

equação do 2º grau

a=1

b=-30

c=125

Δ=b²-4ac

Δ=(-30)²-4(1)(125)

Δ=900-500

Δ=400

$y={-b\pm\sqrt{\Delta} \over2a}={-(-30)\pm\sqrt{400} \over2(1)}={30\pm20\over2}\\ \\y'={30+20\over2}={50\over2}=25 \\ \\ y"={30-20\over2}={10\over2}=5$

$Se~~5^x=y\\ \\ para~~y=25\\ \\ 5^x=25\\ \\ 5^x=5^2\\ \\\fbox{ x=2 }\\ \\ para~~y=5\\ \\ 5^x=5^1\\ \\\fbox{ x=1 }\\ \\ S=\{1,2\}$