Question

Which statement describes if there is an extraneous solution to the equation Vx -3 =x-5? A. No the valid solutions are x= 7 and x=4 B. Yes the extraneous solution is x=4 C. There are no solutions to the equation D. Yes the extraneous solution is x=7

Answer 1

B

Explanation:

given

`√(x-3)` = x - 5 ( suare both sides )

(
`√(x-3)` )² = (x - 5)² ← expand using FOIL

x - 3 = x² - 10x + 25 ( subtract x - 3 from 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 the values of x into the equation and if both sides are equal then it is a solution

x = 4

left side =
`√(4-3)` =
`√(1)` = 1

right side = 4 - 5 = - 1

since 1 ≠ - 1 , then x = 4 is an extraneous solution

x = 7

left side =
`√(7-3)` =
`√(4)` = 2

right side = 7 - 5 = 2

then x = 7 is the solution