Question

√(x+2)+4=x

Answer 1

Solve for x over the real numbers:
4+sqrt(x+2) = x
Isolate the radical to the left hand side.
Subtract 4 from both sides:
sqrt(x+2) = x-4
Eliminate the square root on the left hand side.
Raise both sides to the power of two:
x+2 = (x-4)^2
Write the quadratic polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
x+2 = x^2-8 x+16
Move everything to the left hand side.
Subtract x^2-8 x+16 from both sides:
-x^2+9 x-14 = 0
Factor the left hand side.
The left hand side factors into a product with three terms:
-((x-7) (x-2)) = 0
Multiply both sides by a constant to simplify the equation.
Multiply both sides by -1:
(x-7) (x-2) = 0
Solve each term in the product separately.
Split into two equations:
x-7 = 0 or x-2 = 0
Look at the first equation: Solve for x.
Add 7 to both sides:
x = 7 or x-2 = 0
Look at the second equation: Solve for x.
Add 2 to both sides:
x = 7 or x = 2
Now test that these solutions are correct by substituting into the original equation.
Check the solution x = 2.
sqrt(x+2)+4 => sqrt(2+2)+4 = 6
x => 2 = 2:
So this solution is incorrect
Check the solution x = 7.
sqrt(x+2)+4 => sqrt(2+7)+4 = 7
x => 7 = 7:
So this solution is correct
Gather any correct solutions.
The solution is:
x = 7