Question

1.Sa se rezolve in multimea numerelor complexe ecuatia: x la puterea a doua-3x-4=0

2. Sa se rezolve in multimea nr complexe ecuatia: x supra x+1 + 3 supra x+2 =2
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Answer 1

Răspuns:

$1) {x}^{2} - 3x - 4 = 0 \\ a = 1 \: b = - 3 \: c = - 4 \\ delta = {b}^{2} - 4ac = {( - 3)}^{2} - 4 \times 1 \times ( - 4) = 9 + 16 = 25 \\ delta \geqslant 0 \: rezulta \: ca \: ecuatia \: are \: solutii \: reale \: deci \: nu \: are \: solutii \: complexe$

$2) \frac{x}{x + 1} + \frac{3}{x + 2} = 2 \\ \frac{x(x + 2) + 3(x + 1)}{(x + 1)(x + 2)} = \frac{2(x + 1)(x + 2)}{( x + 1)(x + 2)} \\ x(x + 2) + 3(x + 1) = 2(x + 1)(x + 2) \\ {x}^{2} + 2x + 3x + 3 = 2 \times ( {x}^{2} + 2x + x + 2) \\ {x}^{2} + 5x + 3 = 2( {x}^{2} + 3x + 2) \\ {x}^{2} + 5x + 3 = 2 {x}^{2} + 6x + 4 \\ 0 = 2 {x}^{2} - {x}^{2} + 6x - 5x + 4 - 3 \\ {x}^{2} + x + 1 = 0 \\ a = 1 \: b = 1 \: c = 1 \\ delta = {b }^{2} - 4ac = {1}^{2} - 4 \times 1 \times 1 = 1 - 4 = - 3 \\ x1 = \frac{ - b + i \sqrt{ - delta} }{2a} \\ x1 = \frac{ - 1 + i \sqrt{3} }{2} \\ x2 = \frac{ - b - i \sqrt{ - delta} }{2a} = \frac{ - 1 - i \sqrt{3} }{2}$