Question

Ma poate ajuta cineva cu exercitiul 3?

Answer 1

Răspuns:

$3) {2}^{1 + x} + {2}^{ 1 - x} = 5 \\ 2 \times {2}^{x} + 2 \times {2}^{ - x} = 5 \\ 2 \times {2}^{x} + 2 \times \frac{1}{ {2}^{x} } = 5 \\ notam \: {2}^{x} cu \: t \\ 2 \times t + 2 \times \frac{1}{t} = 5 \\ 2t + \frac{2}{t} = 5 \\ \frac{2t \times t}{t} + \frac{2}{t} = \frac{5t}{t} \\ 2 {t}^{2} + 2 = 5t \\ 2 {t}^{2} - 5t + 2 = 0 \\ a = 2 \\ b = - 5 \\ c = 2 \\ delta \: = {b}^{2} - 4ac = {( - 5)}^{2} - 4 \times 2 \times 2 = 25 - 16 = 9 \\ t1 = \frac{ - b + \sqrt{delta} }{2a} = \frac{ 5 + \sqrt{9} }{2 \times 2} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \\ cum \: {2}^{x } = t \\ {2}^{x} = 2 \\ x1 = 1 \\ t2 = \frac{ - b - \sqrt{delta} }{2a} = \frac{ 5 - \sqrt{9} }{2 \times 2} = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2} \\ cum \: {2}^{x} = t \\ {2}^{x} = \frac{1}{2} \\ {2}^{x} = {2}^{ - 1} \\ x2 = - 1$

S:{-1;2}

S este mulțimea soluțiilor, pune simbolul pt delta