Solve root 5x-1=1+root x+2

Question

Solve root 5x-1=1+root x+2

Answer 1

√(5x - 1) = 1 + √(x + 2)

√(5x - 1) - √(x + 2) = 1

[√(5x - 1) - √(x + 2)]² = 1²

[√(5x - 1)]² - 2√(5x - 1) * √(x + 2) + [√(x + 2)]² = 1 

5x - 1 - 2√[(5x - 1)(x + 2)] + x + 2 = 1

- 2√[(5x - 1)(x + 2)] + 6x + 1 = 1 

- 2√[(5x - 1)(x + 2)] = - 6x - 1 + 1 

- 2√[(5x - 1)(x + 2)] = - 6x 

(√[(5x - 1)(x + 2)])² = (3x)²

(5x - 1)(x + 2) = 9x²

5x² + 9x - 2 - 9x² = 0

- 4x² + 9x - 2 = 0
4x² - 9x + 2 = 0

D = b² - 4ac = 81 - 4*4*2 = 49 

x₁₂ = (- b ± √D) / 2a = (9 ±√49) / 8

x₁ = (9 + 7) / 8 = 2

x₂ = (9 - 7) / 8 = 1/4

Let's check solutions in the original equation 

√(5 * 2 - 1) = 1 + √(2 + 2) 
        √9 = 1 + √4 
          3 = 3

√(5 * 1/4 - 1) = 1 + √(1/4 + 2) 

√(1/4) = 1 + √(9/4)

1/2 ≠ 1 + 3/2

Thus, x = 1/4 is an extraneous root.