Question

Si a+b=6yab=4 calcula a3+b3

Answer 1

Respuesta:

b₁ = 3 + √5

a₁ =  4 / (3 + √5)

a³ + b³ = 144

b₂ = 3 - √5

a₂ =  4 / (3 - √5)

a³ + b³ = 144

Explicación paso a paso:

a + b = 6

ab = 4

ab = 4

a =  4 / b

4 / b  + b = 6

(4 + b²) / b = 6

4 + b² = 6b

b² - 6b + 4 = 0                 b = x

x² - 6x + 4 = 0

x = (-b ± √(b² - 4ac)) / 2a

x = (-(-6) ± √((-6)² - 4(1)(4))) / 2(1)

x = (6 ± √(36 - 16)) / 2

x = (6 ± √(20)) / 2

x = (6 ± √(5*4)) / 2

x = (6 ± 2√5)) / 2

x = 2(3 ± √5)) / 2

x = 3 ± √5

b₁ = 3 + √5

b₂ = 3 - √5

b₁ = 3 + √5

a₁ =  4 / (3 + √5)

a³ + b³

(4 / (3 + √5))³ + (3 + √5)³

(64/(3³ + 3(3)²(√5) + 3(3)(√5)² + (√5)³)) + (3³ + 3(3)²(√5) + 3(3)(√5)² + (√5)³)

(64 / (27 + 27√5 + 9(5) + 5√5)) + (27 + 27√5 + 9(5) + 5√5)

(64 / (27 + 32√5 + 45)) + (27 + 32√5 + 45)

(64 / (72 + 32√5)) + (72 + 32√5)

(64 + (72 + 32√5)²) / (72 + 32√5)

(64 + 5184 + 2(72)(32√5) + (32√5)² )) / (72 + 32√5)

(64 + 5184 + 4608√5 + 1024*5) / (72 + 32√5)

(64 + 5184 + 4608√5 + 5120) / (72 + 32√5)

(10368 + 4608√5) / (72 + 32√5)

144(72 + 32√5) / (72 + 32√5)

144

b₂ = 3 - √5

a₂ =  4 / (3 - √5)

(4 / (3 - √5))³ + (3 - √5)³

(64/(3³ + 3(3)²(-√5)+3(3)(-√5)²+(-√5)³)) + (3³ + 3(3)²(-√5)+3(3)(-√5)²+(-√5)³)

(64 / (27 - 27√5 + 9(5) - 5√5)) + (27 - 27√5 + 9(5) - 5√5)

(64 / (27 - 32√5 + 45)) + (27 - 32√5 + 45)

(64 / (72 - 32√5)) + (72 - 32√5)

(64 + (72 - 32√5)²) / (72 - 32√5)

(64 + 5184 + 2(72)(-32√5) + (-32√5)² )) / (72 - 32√5)

(64 + 5184 - 4608√5 + 1024*5) / (72 - 32√5)

(64 + 5184 - 4608√5 + 5120) / (72 - 32√5)

(10368 - 4608√5) / (72 + 32√5)

144(72 - 32√5) / (72 - 32√5)

144