Question

qual o valor da incógnita x para essa equação:

Answer 1

$4^{x+1}+4^{3-x}=257\\(4^{x}.4^{1})+(4^{3}/4^{x})=257\\4.4^{x}+(64/4^{x})=257$

Chamando 4^x de y:

$4.y+(64/y)=257\\4y+(64/y)=257$

Multiplicando a equação por y:

$y.4y+y(64/y)=257y\\4y^{2}+64=257y\\4y^{2}-257y+64=0$

$\Delta=b^{2}-4.a.c\\\Delta=(-257)^{2}-4*4*64\\\Delta=66049-1024\\\Delta=65025\\\sqrt{\Delta}=\sqrt{65025}\\\sqrt{\Delta}=255$

$y=(-b\pm\sqrt{\Delta})/2a\\y=(-[-257]\pm255)/(2*4)\\y=(257\pm255)/8$

$y'=(257+255)/8\\y'=512/8\\y'=64\\\\y''=(257-255)/8\\y''=2/8\\y''=1/4$
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Como y = 4^x:

$y = 64\\4^{x}=64\\4^{x}=4^{3}\\x=3$

$y=1/4\\4^{x}=1/4\\4^{x}=4^{-1}\\x=-1$

S = {-1,3}