Question

Which statement describes if there is an extraneous solution to the equation √x-3= x - 5?

Yes, the extraneous solution is x = 7.
No, the valid solutions are = 7 and x = 4.
There are no solutions to the equation.
Yes, the extraneous solution is x = 4.
help me please..........

Answer 1

Yes, the extraneous solution is x = 4

Explanation:

given

`√(x-3)` = x - 5 ( square both sides to clear the radical )

(
`√(x-3)` )² = (x - 5)² , that is

x - 3 = (x - 5)² ← expand using FOIL

x - 3 = x² - 10x + 25 (subtract x from both sides )

- 3 = x² - 11x + 25 ( add 3 to both sides )

0 = x² - 11x + 28 ← in standard form

0 = (x - 4)(x - 7) ← in factored form

equate each factor to zero and solve for x

x - 4 = 0 ⇒ x = 4

x - 7 = 0 ⇒ x = 7

As a check

substitute these values for x into the equation and if both sides areequal then they are solutions

x = 4

left side =
`√(x-3)` =
`√(4-3)` =
`√(1)` = 1

right side = x - 5 = 4 - 5 = - 1

since left side ≠ right side then x = 4 is an extraneous solution

x = 7

left side =
`√(x-3)` =
`√(7-3)` =
`√(4)` = 2

right side = x - 5 = 7 - 5 = 2

since left side = right side then x = 7 is the solution