Question

Rezolvati in multimea numerelor reale ecuatia: 2^1+x + 2^1-x =5 va rog mult

Answer 1

${2}^{1 + x} + {2}^{1 - x} = 5$

$2 \times {2}^{x} + 2 \times {2}^{ - x} = 5$

$2( {2}^{x} + {2}^{ - x}) = 5$

${2}^{x} + {2}^{ - x} = \frac{5}{2}$

${2}^{x} + \frac{1}{ {2}^{x} } = \frac{5}{2}$

${2}^{x} + \frac{1}{ {2}^{x} } - \frac{5}{2} = 0$

$notam \: {2}^{x} \: cu \: un \: t$

${2}^{x} = t$

$t + \frac{1}{t} - \frac{5}{2} = 0 \: | \times t$

${t}^{2} + 1 - \frac{5t}{2} = 0 \: | \times 2$

$2 {t}^{2} + 2 - 5t = 0$

$2 {t}^{2} - 5t + 2 = 0$

$a = 2$

$b = - 5$

$c = 2$

$delta = {b}^{2} - 4ac$

$delta = {( - 5)}^{2} - 4 \times 2 \times 2$

$delta = 25 - 16$

$delta = 9$

$t(1.2) = \frac{ - b + - \sqrt{delta} }{2a}$

$t(1.2) = \frac{ - ( - 5) + - \sqrt{9} }{2 \times 2}$

$t(1.2) = \frac{5 + - 3}{4}$

$t(1) = \frac{5 + 3}{4} = \frac{8}{4} = 2$

$t(2) = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2}$

${2}^{x}=2$

$x1=1$

${2}^{x}=\frac{1}{2}$

$x2=-1$