Verified answer

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(2x + 6) - 2 * sqrt(2x + 6) * sqrt(x - 1) + (x - 1) = 4

3x + 5 - 4 = 2 * sqrt(2x + 6) * sqrt(x - 1)

3x + 1 = 2 * sqrt(2x + 6) * sqrt(x - 1)

9x^2 + 6x + 1 = 4 * (2x + 6) * (x - 1)

9x^2 + 6x + 1 = 4 * (2x^2 - 2x + 6x - 6)

9x^2 + 6x + 1 = 4 * (2x^2 + 4x - 6)

9x^2 + 6x + 1 = 8x^2 + 16x - 24

9x^2 - 8x^2 + 6x - 16x + 1 + 24 = 0

x^2 - 10x + 25 = 0

(x - 5) * (x - 5) = 0

x = 5

√(2x+6)-√(x-1) = 2

√(2x+6) = 2-√(x-1)

√(2x+6)^2 = [2-√(x-1)]^2

2x+6 = [2-√(x-1)][2-√(x-1)]

2x+6 = 4-2√(x-1)-2√(x-1)+(x-1)

2x+6 = 4-4√(x-1)+x-1

2x+6 = 3-4√(x-1)+x

2x+6-x-3 = -4√(x-1)

x+3 = -4√(x-1)

(x+3)/(-4) = √(x-1)

[(x+3)/(-4)]^2 = √(x-1)^2

[(x+3)/(-4)][(x+3)/(-4)] = x-1

(x+3)(x+3)/16 = x-1

(x^2+3x+3x+9)/16 = x-1

(x^2+6x+9)/16 = x-1

x^2+6x+9 = 16(x-1)

x^2+6x+9 = 16x-16

x^2+6x+9-16x+16 = 0

x^2-10x+25 = 0

(x-5)(x-5) = 0

x-5 = 0

x=0+5

x=5

√(2x+6) - √(x-1) = 2

Square both sides:

√(2x+6)√(2x+6) - √(2x+6)√(x-1) - √(x-1)√(2x+6) + √(x-1)√(x-1) = 2²

(2x+6) - 2√(2x+6)√(x-1) + (x-1) = 4

3x + 5 - 2√(2x+6)√(x-1) = 4

3x + 1 = 2√(2x+6)√(x-1)

Square both sides again:

(3x + 1)² = 4(2x+6)(x - 1)

9x² + 6x + 1 = 4 (2x² + 4x - 6)

9x² + 6x + 1 = 8x² + 16x - 24

x² - 10x + 25 = 0

(x - 5)² = 0

x = 5

--------------------

Check:

√(2x+6) - √(x-1) = √(10+6) - √(5-1) = √16 - √4 = 4 - 2 = 2 -----> ok

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