Question

Observe uma maneira de resolver a equação (x+2)²=9
De maneira parecida, resolva:
COM CALCULOS.

A) (x+1)²= 16

B) (x-5)²= 4

C) (3+x)²= 49

D) 25= (4-x)²​

Answer 1

Explicação passo-a-passo:

$a) \: \: \: {(x + 1)}^{2} = 16 \\ (x + 1) \times (x + 1) = 16 \\ {x}^{2} + x + x + 1 = 16 \\ {x}^{2} + x + x + 1 - 16 = 0 \\ {x }^{2} + 2x - 15 = 0 \\ \\ a = 1 \\ b = 2 \\ c = - 15 \\ \\ x = \frac{ - b + - \sqrt{ {b}^{2} - 4ac} }{2a} \\ x = \frac{ - 2 + - \sqrt{ {2}^{2} - 4 \times 1 \times ( - 15)} }{2 \times 1} \\ x = \frac{ - 2 + - \sqrt{4 + 60} }{2} \\ x = \frac{ - 2 + - \sqrt{64} }{2} \\ x = \frac{ - 2 + - 8}{2} \\ \\ {x}^{1} = \frac{ - 2 + 8}{2} = {x}^{1} = \frac{6}{2} = {x}^{1} = 3 \\ {x}^{2} = \frac{ - 2 - 8}{2} = {x}^{2} = - \frac{10}{2} = {x}^{2} = - 5$

$b) \: \: \: {(x - 5)}^{2} = 4 \\ (x - 5) \times (x - 5) = 4 \\ {x}^{2} - 5x - 5x + 25 = 4 \\ {x}^{2} - 5x - 5x + 25 - 4 = 0 \\ {x }^{2} - 10x + 21 = 0 \\ \\ a = 1 \\ b = - 10 \\ c = 21 \\ \\ x = \frac{ - ( - 10) + - \sqrt{ {( - 10)}^{2} - 4 \times 1 \times 21} }{2 \times 1} \\ x = \frac{10 + - \sqrt{100 - 84} }{2} \\ x = \frac{10 + - \sqrt{16} }{2} \\ x = \frac{10 + - 4}{2} \\ \\ {x}^{1} = \frac{10 + 4}{2 } = {x}^{1} = \frac{14}{2} = {x}^{1} = 7 \\ {x}^{2} = \frac{10 - 4}{2} = {x}^{2} = \frac{6}{2} = {x}^{2} = 3$

$c) \: \: \: {(3 + x)}^{2} = 49 \\ (3 + x) \times (3 + x) = 49 \\ 9 + 3x + 3x + {x}^{2} = 49 \\ {x}^{2} + 3x + 3x + 9 - 49 = 0 \\ {x}^{2} + 6x - 40 = 0 \\ \\ a = 1 \\ b = 6 \\ c = - 40 \\ \\ x = \frac{ - 6 + - \sqrt{ {6}^{2} - 4 \times 1 \times ( - 40)} }{2 \times 1} \\ x = \frac{ - 6 + - \sqrt{36 + 160} }{2} \\ x = \frac{ - 6 + - \sqrt{196} }{2} \\ x = \frac{ - 6 + - 14}{2} \\ \\ {x}^{1} = \frac{ - 6 + 14}{2} = {x}^{1} = \frac{8}{2} = {x}^{1} = 4 \\ {x}^{2} = \frac{ - 6 - 14}{2} = {x}^{2} = - \frac{20}{2} = {x}^{2} = - 10$

$d) \: \: \: 25 = {(4 - x)}^{2} \\ {(4 - x)}^{2} = 25 \\ (4 - x) \times (4 - x) = 25 \\ 16 - 4x - 4x + {x}^{2} = 25 \\ {x}^{2} - 4x - 4x + 16 - 25 = 0 \\ {x}^{2} - 8x - 9 = 0 \\ \\ a = 1 \\ b = - 8 \\ c = - 9 \\ \\ x = \frac{ - ( - 8) + - \sqrt{ {( - 8)}^{2} - 4 \times 1 \times ( - 9)} }{2 \times 1} \\ x = \frac{8 + - \sqrt{64 + 36} }{2} \\ x = \frac{8 + - \sqrt{100} }{2} \\ x = \frac{8 + - 10}{2} \\ \\ {x}^{1} = \frac{8 + 10}{2} = {x}^{1} = \frac{18}{2} = {x}^{1} = 9 \\ {x}^{2} = \frac{8 - 10}{2} = {x}^{2} = - \frac{2}{2} = {x}^{2} = - 1$