Question

Equação do segundo grau. Resolva:
a) 5x² - 10x = 0
b) 3x² - 7x = x (2x - 4)
c) 9x² = 54x
d) (x-5) (x-6) = 30
e) x (x+2) = 2x + 25

Answer 1

Equação do segundo grau. Resolva:
a) 5x² - 10x = 0   equação INCOMPLETA podemos fazer


  5x² - 10x = 0
5x(x - 2)= 0

5x = 0
x = 0/5
x = 0
e
(x - 2) = 0
x - 2 = 0
x = + 2
assim
x' = 0 
x" = 2

b) 3x² - 7x = x (2x - 4)
    3x² - 7x = 2x² - 4x   ( igualar a ZERO)
     3x² - 2x² - 7x + 4x = 0
1x² - 3x = 0

1x² - 3x = 0
1x(x - 3) =0
1x = 0
x = 0/1
x = 0
e
(x - 3) = 0
x -  3 = 0
x = + 3

assim
x' = 0
x" = 3 





c) 9x² = 54x

9x² - 54x = 0
9x(x - 6) = 0
9x = 0
x = 0/9
x = 0
e
(x - 6) = 0
x - 6 = 0
x = + 6

assim
x' = 0
x" = 6




d) (x-5) (x-6) = 30
x² - 6x - 5x + 30 = 30
x² - 11x + 30 = 30
x² - 11x  + 30 - 30 = 0
x² - 11x        0        = 0

x² - 11x = 0
x(x - 11) = 0 
x = 0
e
(x - 11)= 0
x - 11 = 0
x = + 11

assim
x'  0
x" = 11

e) x (x+2) = 2x + 25
    x² + 2x = 2x + 25
x² + 2x - 2x = 25
x²         0     = 25

x² = 25
x = + - √25       (√25 = 5)
x = + - 5

assim
x' = - 5
x" = + 5

Answer 2

Oie, Td Bom?!

a)

$5x {}^{2} - 10x = 0 \\ 5x \: . \: (x - 2) = 0 \\ x \: . \: (x - 2) = 0$

$x = 0 \\ x - 2 = 0⟶x = 2$

S = $\left \{ x_{1} = 0 \: , \: x_{2} = 2 \right \}$

b)

$3x {}^{2} - 7x = x \: . \: (2x - 4) \\ 3x {}^{2} - 7x = 2x {}^{2} - 4x \\ 3x { }^{2} - 7x - 2x {}^{2} + 4x = 0 \\ x {}^{2} - 3x = 0 \\ x \: . \: (x - 3) = 0$

$x = 0 \\ x - 3 = 0⟶x = 3$

S = $\left \{ x_{1} = 0 \: , \: x_{2} = 3 \right \}$

c)

$9x {}^{2} = 54x \\ x {}^{2} = 6x \\ x {}^{2} - 6 = 0 \\ x \: . \: (x - 6) = 0$

$x = 0 \\ x - 6 = 0⟶x = 6$

S = $\left \{ x_{1} = 0 \: , \: x_{2} = 6 \right \}$

d)

$(x - 5) \: . \: (x - 6) = 30 \\ x {}^{2} - 6x - 5x + 30 = 30 \\ x {}^{2} - 6x - 5x = 0 \\ x {}^{2} - 11x = 0 \\ x \: . \: (x - 11) = 0$

$x = 0 \\ x - 11 =0⟶x = 11$

S = $\left \{ x_{1} = 0 \: , \: x_{2} = 11 \right \}$

e)

$x \: . \: (x + 2) = 2x + 25 \\ x {}^{2} + 2x = 2x + 25 \\ x {}^{2} = 25 \\ x {}^{2} - 25 = 0 \\ (x - 5) \: . \: (x + 5) = 0$

$x - 5 = 0⟶ x= 5 \\ x + 5 = 0⟶x = - 5$

S = $\left \{ x_{1} = - 5 \: , \: x_{2} = 5 \right \}$

Att. Makaveli1996