Verified answer

√(2x + 6) - √(x + 4) = 1

√(2x + 6) = √(x + 4) + 1

square both sides

2x + 6 = (x + 4) + 1 + 2√(x + 4)

2x + 6 = x + 5 + 2√(x + 4)

x + 1 = 2√(x + 4)

square both sides

x² + 2x + 1 = 4(x + 4)

x² + 2x + 1 = 4x + 16

x² - 2x - 15 = 0

By the quadratic formula,

x = [2 ± √(2² – 4·1(-15))] / [2·1]

   = [2 ± √64] / 2

   = [2 ± 8] /2

   = -3, 5

√(2(-3) + 6) - √(-3 + 4) = -1, so x ≠ -3

√(2·5 + 6) - √(5 + 4) = 1, so x = 5

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Note that Rebecca's answer is incorrect. Squaring both sides does NOT give you (2x+6)-(x+4)=1

√(2x+6)-√(x+4)=1

get it so that there are no minuses in the equation

(squaring destroys it so that's why you need to save it by moving it to the other side)

√(2x+6)=1+√(x+4)

square everything

2x+6=1+2√(x+4)+x+4

move all the things to the left leaving the √ alone

2x+6-1-x-4 = 2√(x+4)

evaluate left side

x-1 = 2√(x+4)

square again

x² - 2x + 1 = 4*(x+4)

evaluate

x² - 2x +1 = 4x - 4

x² - 2x +1 - 4x + 4 = 0

x² - 6x + 5 = 0

use the quadratic equation to solve this.

x₁₂ = [6±√(36-4*1*5)]/2

x₁₂ = [6±√(36-20)]/2

x₁₂ = [6±√(16)]/2

x₁₂ = [6±4]/2

x₁₂ = 3±2

Solutions:

x₁ = 5

x₂ = 1

Test (Required because conditions are omitted):

√(2*5+6)-√(5+4)=1

√(16)-√(9)=1

4-3 = 1

CORRECT!

√(2*1+6)-√(1+4)=1

√(8)-√(5)=1

INCORRECT!

So the final solution is: x = 5

Squaring both sides:

(2x + 6) - (x + 4) - 2√(2x + 6)√(x + 4) = 1

x + 2 - 2√(2x + 6)√(x + 4) = 1

-2√(2x + 6)√(x + 4) = -1 - x

(2x + 6)(x + 4) = (1 + 2x + x²)/4

8x^2 + 56x + 96 = 1 + 2x + x^2

7x^2 + 54x + 95 = 0

x = (-54 +/- √(54^2 - 4(7)(95))/14

x = -2.714, -5

Note that x = -5 is a solution that's not valid since it leads to a negative inside both radicals.

Answer => x = -2.714

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square everything

then you get

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

now just solve