Question

Resolva as equações completas no conjunto R: a) 2x2 + 7x + 5 = 0 b) x2 + 2x − 8 = 0 c) −x2 + 10x − 24 = 0 d) −x2 − 2x + 3 = 0 e) y2 + 4y + 4 = 0 f) 3x2 − 8x + 10 = 0 g) 8x2 − 2x − 1 = 0

Answer 1

Oie, Td Bom?!

a)

$2x {}^{2} + 7x + 5 = 0$

$2x {}^{2} + 5x + 2x + 5 = 0$

$x \: . \: (2x + 5) + 2x + 5 = 0$

$(2x + 5) \: . \: (x + 1) = 0$

• $2x + 5 = 0⇒x = - \frac{5}{2}$

• $x + 1 = 0⇒x = - 1$

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

b)

$x {}^{2} + 2x - 8 = 0$

$x {}^{2} + 4x - 2x - 8 = 0$

$x \: . \: (x + 4) - 2(x + 4) = 0$

$(x + 4) \: . \: (x - 2) = 0$

• $x + 4 = 0⇒x = - 4$

• $x - 2 = 0⇒x = 2$

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

c)

$- x {}^{2} + 10x - 24 = 0$

$x {}^{2} - 10x + 24 = 0$

$x {}^{2} - 4x - 6x + 24 = 0$

$x \: . \: (x - 4) - 6(x - 4) = 0$

$(x - 4) \: . \: (x - 6) = 0$

• $x - 4 = 0⇒x = 4$

• $x - 6 = 0⇒x = 6$

$S = \left \{ 4 \: , \: 6\right \}$

d)

$- x {}^{2} - 2x + 3 = 0$

$x {}^{2} + 2x - 3 = 0$

$x {}^{2} + 3x - x - 3 = 0$

$x \: . \: (x + 3) - (x + 3) = 0$

$(x + 3) \: . \: (x - 1) = 0$

• $x + 3 = 0⇒x = - 3$

• $x - 1 = 0⇒x = 1$

$S = \left \{ - 3 \: ,\: 1\right \}$

e)

$y {}^{2} + 4y + 4 = 0$

$y {}^{2} + 2 \: . \: y \: . \: 2 + 2 {}^{2} = 0$

$(y + 2) {}^{2} = 0$

$y + 2 = 0$

$y = - 2$

$S = \left \{ - 2\right \}$

f)

$3x {}^{2} - 8x + 10 = 0$

• Coeficientes:

$a = 3 \: , \: b = - 8 \: , \: c = 10$

• Fórmula resolutiva:

$x = \frac{ - b± \sqrt{b {}^{2} - 4ac} }{2a}$

$x= \frac{ -( - 8)± \sqrt{( - 8) {}^{2} - 4 \: . \: 3 \: . \: 10 } }{2 \: . \: 3}$

$x = \frac{8± \sqrt{64 - 120} }{6}$

$x = \frac{8± \sqrt{ - 56} }{6}$

• A raiz quadrada do número negativo não pertence ao intervalo dos Números Reais.

$x∉\mathbb{R}$

g)

$8x {}^{2} - 2x - 1 = 0$

$8x {}^{2} + 2x - 4x - 1 = 0$

$2x \: . \: (4x + 1) - (4x + 1) = 0$

$(4x + 1) \: . \: (2x - 1) = 0$

• $4x + 1 = 0⇒ x = - \frac{1}{4}$

• $2x - 1 = 0⇒x = \frac{1}{2}$

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

Att. Makaveli1996

Answer 2

Explicação passo-a-passo:

a) 2x² + 7x + 5 = 0

Δ = 7² - 4.2.5

Δ = 49 - 40

Δ = 9

x = (-7 ± √9)/2.2 = (-7 ± 3)/4

• x' = (-7 + 3)/4 = -4/4 = -1

• x" = (-7 - 3)/4 = -10/4 = -5/2

b) x² + 2x - 8 = 0

Δ = 2² - 4.1.(-8)

Δ = 4 + 32

Δ = 36

x = (-2 ± √36)/2.1 = (-2 ± 6)/2

• x' = (-2 + 6)/2 = 4/2 = 2

• x" = (-2 - 6)/2 = -8/2 = -4

c) -x² + 10x - 24 = 0

Δ = 10² - 4.(-1).(-24)

Δ = 100 - 96

Δ = 4

x = (-10 ± √4)/2.(-1) = (-10 ± 2)/-2

• x' = (-10 + 2)/-2 = -8/-2 = 4

• x" = (-10 - 2)/-2 = -12/-2 = 6

d) -x² - 2x + 3 = 0

Δ = (-2)² - 4.(-1).3

Δ = 4 + 12

Δ = 16

x = (2 ± √16)/2.(-1) = (2 ± 4)/-2

• x' = (2 + 4)/-2 = 6/-2 = -3

• x" = (2 - 4)/-2 = -2/-2 = 1

e) y² + 4y + 4 = 0

Δ = 4² - 4.1.4

Δ = 16 - 16

Δ = 0

y = (-4 ± √0)/2.1 = (-4 ± 0)/2

• y' = y" = -4/2

y' = y" = -2

f) 3x² - 8x + 10 = 0

Δ = (-8)² - 4.3.10

Δ = 64 - 120

Δ = -56

Como Δ < 0, não há raízes reais

g) 8x² - 2x - 1 = 0

Δ = (-2)² - 4.8.(-1)

Δ = 4 + 32

Δ = 36

x = (2 ± √36)/2.8 = (2 ± 6)/16

• x' = (2 + 6)/16 = 8/16 = 1/2

• x" = (2 - 6)/16 = -4/16 = -1/4