Question

Worth 30 points.

Show all work to solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.

the square root of x minus 3 plus five equals x.

Picture of question and equation attached.

Answer 1

x=7 valid solution

x=4 extraneous solution

Explanation:

sqrt( x-3) +5 = x

Subtract 5 from each side

sqrt( x-3) +5 -5= x-5

sqrt( x-3) = x-5

Square each side

(sqrt( x-3))^2 = (x-5)^2

Foil the right side ( x-5)(x-5) =x^2 -5x-5x+25 = x^2 -10x+25

x-3 = x^2 -10x+25

Subtract x from each side

-3 = x^2 -10x -x+25

Add 3 to each side

0 = x^2 -11x +25 +3

0 = x^2 -11x+28

Factor

0 = (x-7)(x-4)

Using the zero product property

x-7 =0 x-4=0

x=7 x=4

Check the solutions

sqrt( x-3) +5 = x

x=7

sqrt( 7-3) +5 = 7

sqrt(4) +5 = 7

2 +5 = 7 valid solution

x=4

sqrt( 4-3) +5 = 4

sqrt(1) +5 = 4

1 +5 = 4 not true extraneous solution