Question

2 - Faça a fatoração e encontre as raízes das equações: a) -3x² + 9x = 0 b) 2x² + 3x = 0 c) (x+5)(x-1) = 2x - 5 d) x 2 + 3x + 2 = 0 e) x 2 + 6x + 8 = 0 f) x 2 - 2x - 15 = 0 g) (x + 1)(x - 3) = x + 7

Answer 1

Explicação passo-a-passo:

a) $\sf -3x^2+9x=0$

$\sf -3x\cdot(x-3)=0$

$\sf -3x=0~\rightarrow~x'=0$

$\sf x-3=0~\rightarrow~x"=3$

b) $\sf 2x^2+3x=0$

$\sf x\cdot(2x+3)=0$

$\sf x'=0$

$\sf 2x+3=0~\rightarrow~x"=\dfrac{-3}{2}$

c) $\sf (x+5)\cdot(x-1)=2x-5$

$\sf x^2-x+5x-5-2x+5=0$

$\sf x^2+2x=0$

$\sf x\cdot(x+2)=0$

$\sf x'=0$

$\sf x+2=0~\rightarrow~x"=-2$

d) $\sf x^2+3x+2=0$

$\sf x^2+x+2x+2=0$

$\sf x\cdot(x+1)+2\cdot(x+1)=0$

$\sf (x+1)\cdot(x+2)=0$

$\sf x+1=0~\rightarrow~x'=-1$

$\sf x+2=0~\rightarrow~x"=-2$

e) $\sf x^2+6x+8=0$

$\sf x^2+2x+4x+8=0$

$\sf x\cdot(x+2)+4\cdot(x+2)=0$

$\sf (x+2)\cdot(x+4)=0$

$\sf x+2=0~\rightarrow~x'=-2$

$\sf x+4=0~\rightarrow~x"=-4$

f) $\sf x^2-2x-15=0$

$\sf x^2+3x-5x-15=0$

$\sf x\cdot(x+3)-5\cdot(x+3)=0$

$\sf (x+3)\cdot(x-5)=0$

$\sf x+3=0~\rightarrow~x'=-3$

$\sf x-5=0~\rightarrow~x"=5$

g) $\sf (x+1)\cdot(x-3)=x+7$

$\sf x^2-3x+x-3-x-7=0$

$\sf x^2-3x-10=0$

$\sf x^2+2x-5x-10=0$

$\sf x\cdot(x+2)-5\cdot(x+2)=0$

$\sf (x+2)\cdot(x-5)=0$

$\sf x+2=0~\rightarrow~x'=-2$

$\sf x-5=0~\rightarrow~x"=5$

Answer 2

Oie, Td Bom?!

a)

$- 3x {}^{2} + 9x = 0 \\ - 3x \: . \: (x - 3) = 0 \\ x \: . \: (x - 3) = 0$

$x = 0 \\ x - 3 = 0⟶x = 3$

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

b)

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

$x = 0 \\ 2x + 3 = 0⟶x = - \frac{3}{2}$

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

c)

$(x + 5) \: . \: (x - 1) = 2x - 5 \\ x {}^{2} - x + 5x - 5 = 2x - 5 \\ x{}^{2} - x + 5x = 2x \\ x {}^{2} + 4x = 2x \\ x {}^{2} + 4x - 2x = 0 \\ x {}^{2} + 2x = 0 \\ x \: . \: (x + 2) = 0$

$x = 0 \\ x + 2 = 0⟶x = - 2$

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

d)

$x {}^{2} + 3x + 2 = 0 \\ x {}^{2} + 2x + x + 2 = 0 \\ x \: . \: (x + 2) + x + 2 = 0 \\ (x + 2) \: . \: (x +1 ) = 0$

$x + 2 = 0 ⟶x = - 2\\ x + 1 = 0⟶x = - 1$

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

e)

$x {}^{2} + 6x + 8 = 0 \\ x {}^{2} + 4x + 2x + 8 = 0 \\ x \: . \: (x + 4) + 2(x + 4) = 0 \\ (x + 4) \: . \: (x + 2) = 0$

$x + 4 = 0 ⟶x = - 4\\ x + 2 = 0⟶x = - 2$

$S= \left \{x_{1} = - 4 \: \: x_{2} = - 2 \right \}$

f)

$x {}^{2} - 2x - 15 = 0\\ x {}^{2} + 3x - 5x - 15 = 0 \\ x \: . \: (x + 3) - 5(x + 3) = 0 \\ (x + 3) \: . \: (x - 5) = 0$

$x + 3 = 0⟶x = - 3 \\ x - 5 = 0⟶x = 5$

$S= \left \{ x_{1} = - 3 \: \: x_{2} = 5 \right \}$

g)

$(x + 1) \: . \: (x - 3) = x + 7 \\ x {}^{2} - 3x + x - 3 = x + 7 \\ x {}^{2} - 3x - 3 = 7 \\ x {}^{2} - 3x - 3 - 7 = 0 \\ x {}^{2} + 2x - 5x - 10 = 0 \\ x \: . \: (x + 2) - 5(x + 2) = 0 \\ (x + 2) \: . \: (x - 5) = 0$

$x + 2 = 0⟶x = - 2 \\ x - 5 = 0⟶x = 5$

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

Att. Makaveli1996