Question

Show work and solve for √ ( 5 x − 9 ) − 1 = x.

Identify any extraneous solution. Show work!

(Use sqrt ( ) for square root.)

Answer 1

sqrt (5x - 9) - 1 = x

sqrt(5x - 9) = x + 1

Square both sides:_

5x - 9 = x^2 + 2x + 1

x^2 - 3x + 10 = 0

(x - 5)(x + 2) = 0

x = 5 or -2.

Lets look for for any extraneous solutions:-

x = 5 ; sqrt (5*5-9) - 1 = sqrt16 - 1 = 3 and x = 3 (right hand side of equation)

so x = 5 is a solution

x = -2: sqrt(5*-2 - 9) - 1 = sqrt (-19) - 1 and x = -2 so this is extraneous

Answer:- One solution x = 5

Answer 2

x = -2 and x = 5

Explanation:

`√((5x-9)) -1=x\\\\√((5x-9))=x+1`

Taking square on both sides of the equation to get:

`(√(5x-9))^2=(x+1)^2\\\\5x-9=x^2+2x+1\\\\x^2+2x-5x+1+9=0\\\\x^2-3x+10=0`

Factorizing this equation to get:

`x^2+2x-5x+10=0\\\\x(x+2)-5(x+2)=0\\\\(x+2)(x-5)=0\\\\x=-2, x=5`

Checking for any extraneous solution:

Put x=-2 in the given expression:

`√(5(-2)-9) -1=-2\\\\√(-19)-1 \\eq -2` ---> extraneous solution