Question

Resolva as equações do 2º grau completas utilizando a fórmula geral. a) x² - 5x + 6 = 0 b) x² - 8x + 12 = 0 c) x² + 2x - 8 = 0 d) x² - 5x + 8 = 0 e) 2x² - 8x + 8 = 0 f) x² - 4x - 5 = 0 g) -x² + x + 12 = 0

Answer 1

Resposta:

a) x' = 3; x'' = 2

b) x' = 6; x'' = 2

c) x' = 2; x'' = -4

d) x' = 5 + √-7 / 2, x'' = 5 - √-7 / 2

e) x = 2

f) x' = 5, x'' = -1

g) x' = 4; x'' = -3

Explicação passo-a-passo:

a) x² - 5x + 6 = 0

a = 1; b = -5; c = 6

Δ = b² - 4ac

Δ = (-5)² - 4 * 1 * 6

Δ = 25 - 24

Δ = 1

x = -b ± √Δ / 2a

x = -(-5) ± √1 / 2 * 1

x = 5 ± 1 / 2

x' = 5 + 1 / 2

x' = 6 / 2

x' = 3

x'' = 5 - 1 / 2

x'' = 4 / 2

x'' = 2

S = {x' = 3, x'' = 2}

b) x² - 8x + 12 = 0

a = 1; b = -8; c = 12

Δ = b² - 4ac

Δ = (-8)² - 4 * 1 * 12

Δ = 64 - 48

Δ = 16

x = -b ± √Δ / 2a

x = -(-8) ± √16 / 2 * 1

x = 8 ± 4 / 2

x' = 8 + 4 / 2

x' = 12 / 2

x' = 6

x'' = 8 - 4 / 2

x'' = 4 / 2

x'' = 2

S = {x' = 6, x'' = 2}

c) x² + 2x - 8 = 0

a = 1; b = 2; c = -8

Δ = b² - 4ac

Δ = 2² - 4 * 1 * (-8)

Δ = 4 - (-32)

Δ = 4 + 32

Δ = 36

x = -b ± √Δ / 2a

x = -2 ± √36 / 2 * 1

x = -2 ± 6 / 2

x' = -2 + 6 / 2

x' = 4 / 2

x' = 2

x'' = -2 - 6 / 2

x'' = -8 / 2

x'' = -4

S = {x' = 2, x'' = -4}

d) x² - 5x + 8 = 0

a = 1; b = -5; c = 8

x = -b ± √b² - 4ac / 2a

x = -(-5) ± √(-5)² - 4 * 1 * 8 / 2 * 1

x = 5 ± √25 - 32 / 2

x = 5 ± √-7 / 2

x' = 5 + √-7 / 2

x'' = 5 - √-7 / 2

S = {x' = 5 + √-7 / 2; x'' = 5 - √-7 / 2}

e) 2x² - 8x + 8 = 0

x² - 4x + 4 = 0

a = 1; b = -4; c = 4

Δ = b² - 4ac

Δ = (-4)² - 4 * 1 * 4

Δ = 16 - 16

Δ = 0

Apenas uma solução.

x = -b ± √Δ / 2a

x = -(-4) ± √0 / 2 * 1

x = 4 ± 0 / 2

x = 4 / 2

x = 2

S = {x = 2}

f) x² - 4x - 5 = 0

a = 1; b = -4; c = -5

Δ = b² - 4ac

Δ = (-4)² - 4 * 1 * (-5)

Δ = 16 - (-20)

Δ = 16 + 20

Δ = 36

x = -b ± √Δ / 2a

x = -(-4) ± √36 / 2 * 1

x = 4 ± 6 / 2

x' = 4 + 6 / 2

x' = 10 / 2

x' = 5

x'' = 4 - 6 / 2

x'' = -2 / 2

x'' = -1

S = {x' = 5; x'' = -1}

g) -x² + x + 12 = 0

x² - x - 12 = 0

a = 1; b = -1; c = -12

Δ = b² - 4ac

Δ = (-1)² - 4 * 1 * (-12)

Δ = 1 - (-48)

Δ = 1 + 48

Δ = 49

x = -b ± √Δ / 2a

x = -(-1) ± √49 / 2 * 1

x = 1 ± 7 / 2

x' = 1 + 7 / 2

x' = 8 / 2

x' = 4

x'' = 1 - 7 / 2

x'' = -6 / 2

x'' = -3

S = {x' = 4; x'' = -3}