Answer:
9/2+sqrt(21)/2 = x
Step-by-step explanation:
sqrt(x+1) = x-4
Square each side
(sqrt(x+1))^2 = (x-4)^2
x+1 = (x-4) (x-4)
FOIL
x+1 = x^2 -4x-4x+16
Combine like terms
x+1 = x^2 -8x+16
Subtract x from each side
x-x+1 = x^2 -8x-x+16
1 = x^2 -9x+16
Subtract 16 from each side
1-16 = x^2 -9x+16-16
-15=x^2 -9x
Completing the square
(-9/2)^2 = 81/4
Add 81/4
-15 +81/4 = x^2 -9x +81/4
Get a common denominator
-60/4 +81/4 = (x-9/2)^2
21/4 = (x-9/2)^2
Take the square root of each side
±sqrt(21/4) = sqrt( (x-9/2)^2)
±sqrt(21/4) = (x-9/2)
We know the sqrt(a/b) =sqrt(a)/sqrt(b)
±sqrt(21)/sqrt(4) = (x-9/2)
±sqrt(21)/2 = (x-9/2)
Add 9/2 to each side
9/2±sqrt(21)/2 = x-9/2+9/2
9/2±sqrt(21)/2 = x
We need to check the solutions since we squared as our first step and can get extraneous solutions
sqrt(9/2+sqrt(21)/2+1) = 9/2+sqrt(21)/2-4
This is a valid solution
sqrt(9/2-sqrt(21)/2+1) = 9/2-sqrt(21)/2-4
The right side is negative and the left side is a square root. This cannot happen, so it is an extraneous solution
