Question

Is the solution of this equation extraneous?

Answer 1

x = - 3 is extraneous

Explanation:

Given

`√(x+7)` - 1 = x ( add 1 to both sides )

`√(x+7)` = 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 :
`√(-3+7)` - 1 =
`√(4)` - 1 = 2 - 1 = 1 ≠ - 3

x = 2 :
`√(2+7)` - 1 =
`√(9)` - 1 = 3 - 1 = 2

x = 2 is a solution and x = - 3 is extraneous

Answer 2

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Explanation:

Look at the photo below for the answer.

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