Question

Resolva em R a equação. √(x+7) + √(x-5) = √(2x+18)

Answer 1

$\sqrt{x + 7} = \sqrt{2x + 18} - \sqrt{x - 5} \\ {( \sqrt{x + 7} )}^{2} = {( \sqrt{2x + 18} - \sqrt{x - 5} )}^{2} \\ x + 7 = 2x + 18 - 2 \sqrt{(2x + 18)(x - 5)} + x - 5 \\ x + 7 - 2x - 18 - x + 5 = - 2 \sqrt{(2x + 18)(x - 5)} \\ - 2x - 6 = - 2 \sqrt{(2x + 18)(x - 5)} \\ {( - 2x - 6)}^{2} = {( - 2 \sqrt{(2x + 18)(x - 5)} )}^{2} \\ 4 {x}^{2} + 24x + 36 = 4(2x + 18)(x - 5) \\ 4 {x}^{2} + 24x + 36 = 4(2 {x}^{2} - 10x + 18x - 90) \\ 4 {x}^{2} + 24x + 36 = 4(2 {x}^{2} + 8x - 90) \\ {x }^{2} + 6x + 9 = 2 {x}^{2} + 8x - 90 \\ - {x}^{2} - 2x + 99 = 0 \\ {x}^{2} + 2x - 99 = 0 \\ delta = 4 - 4 \times 1 \times ( - 99) \\ delta = 4 + 396 \\ delta = 400 \\ x 1= \frac{ - 2 + 20}{2} = \frac{18}{2} = 9 \\ x2 = \frac{ - 2 - 20}{2} = - \frac{22}{2} = - 11 \\ \\ verificando \: para \: x = 9 \: confere. \\ verificando \: para \: x = - 11 \: nao \: confere.$

Logo, S = {9}.