Answer:
x = - 3 is extraneous
Step-by-step explanation:
Given
[tex]\sqrt{x+7}[/tex] - 1 = x ( add 1 to both sides )
[tex]\sqrt{x+7}[/tex] = x + 1 ( square both sides )
x + 7 = (x + 1)² ← distribute right side
x + 7 = x² + 2x + 1 ( subtract x + 7 from both sides )
0 = x² + x - 6 ← in standard form
0 = (x + 3)(x - 2) ← in factored form
Equate each factor to zero and solve for x
x + 3 = 0 ⇒ x = - 3
x - 2 = 0 ⇒ x = 2
As a check
Substitute these values into the equation and if both sides are equal then they are the solutions.
x = - 3 : [tex]\sqrt{-3+7}[/tex] - 1 = [tex]\sqrt{4}[/tex] - 1 = 2 - 1 = 1 ≠ - 3
x = 2 : [tex]\sqrt{2+7}[/tex] - 1 = [tex]\sqrt{9}[/tex] - 1 = 3 - 1 = 2
x = 2 is a solution and x = - 3 is extraneous
