Question

Resolva as equações
A) x²-10x+9=0
B) 4x²-4x+1=0
C)x²-4x+8=0
D) x²-2x-3=0

Answer 1

Resposta:

A) x' = 9; x" = 1

B) x = 1/2

C) x' = 2 + 2.i; x" = 2 - 2.i

D) x' = 3; x" = - 1

Explicação passo-a-passo:

A) x²-10x+9=0

∆ = (-10)² - 4.1.9

∆ = 100 - 36 = 64

√∆ = √64 = 8

x = [-(- 10) + 8]/2

x = (10 ± 8)/2.1

x = 5 ±4

x' = 5 + 4 = 9

x" = 5 - 4 = 1

B) 4x²-4x+1=0

x = [4 ± √(-4)² - 4.4.1)]/2.4

x = (4 ± √16 - 16)/8

x = (4 ± 0)/8

x = 4/8 = 1/2

C)x²-4x+8=0

x = [4 ± √(-4)² - 4.8)]/2

x = [4 ± √(16 - 32)]/2

x = [4 ± √(-16)]/2

x = (4 ± √(-16)/2 ← impossível com números reais

Com números imaginários (complexos)

x = [4 ± √-1 × √16]/2

x = (4 ± i.4)/2

x = 2 ± 2.i

x' = 2 + 2.i

x" = 2 - 2.i

i = √-1

D) x²-2x-3=0

x = [2 ± √(2² + 4.1.3)]/2

x = (2 ± √4+12)/2

x = (2 ± √16)/2

x = (2 ± 4)/2

x = 1 ± 2

x' = 1+2 = 3

x" = 1 - 2 = - 1

Answer 2

Resposta:

A) x = 1 e 9.

B) x = 1/2 ou 0,5.

C) x = x∉R.

D) x = (–1) e 3.

Explicação passo a passo:

A) x²–10x+9=0

x = (–b ± √(b² – 4 • a • c)) / (2 • a)

x = (–(–10) ± √((–10)² – 4 • 1 • 9)) / (2 • 1)

x = (10 ± √(100 – 36)) / 2

x = (10 ± √64) / 2

x = (10 ± √8²) / 2

x = (10 ± 8) / 2

x1 = (10 + 8) / 2

x1 = 18 / 2

x1 = 9

x2 = (10 – 8) / 2

x2 = 2 / 2

x2 = 1

B) 4x²–4x+1=0

x = (–b ± √(b² – 4 • a • c)) / (2 • a)

x = (–(–4) ± √((–4)² – 4 • 4 • 1)) / (2 • 4)

x = (4 ± √(16 – 16)) / 8

x = (4 ± √0) / 8

x = (4 ± 0) / 8

x = 4 / 8

x = 1/2 ou 0,5

C)x²–4x+8=0

x = (–b ± √(b² – 4 • a • c)) / (2 • a)

x = (–(–4) ± √((–4)² – 4 • 1 • 8)) / (2 • 1)

x = (4 ± √(16 – 32)) / 2

x = (4 ± √(–16)) / 2

x∉R --> x não existe nos números reais. --> não existe raiz quadrada de número negativo.

D) x²–2x–3=0

x = (–b ± √(b² – 4 • a • c)) / (2 • a)

x = (–(–2) ± √((–2)² – 4 • 1 • (–3))) / (2 • 1)

x = (2 ± √(4 –(–12))) / 2

x = (2 ± √(4 + 12)) / 2

x = (2 ± √16) / 2

x = (2 ± √4²) / 2

x = (2 ± 4) / 2

x1 = (2 + 4) / 2

x1 = 6 / 2

x1 = 3

x2 = (2 – 4) / 2

x2 = (–2) / 2

x2 = (–1)