How do you solve #sqrt( x+5) + 1 = x #?

Question

Nathan Axel · Accepted Answer

#x=4#

We isolate the square root first so that we can simplify by squaring.

First subtract #1# from both sides to get:

#sqrt(x+5) = x - 1#

Next square both sides (which may result in spurious solutions) to get:

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

Subtract #x+5# from both sides to get:

#0 = x^2-3x-4 = (x-4)(x+1)#

Hence #x = 4# or #x = -1#

The solution #x=-1# of this quadratic is not a solution of the original equation:

#sqrt(-1+5) = sqrt(4) = 2 != -2 = -1-1#

The other solution #x=4# is a solution of the original equation:

#sqrt(4+5) = sqrt(9) = 3 = 4 - 1#

Jace Anderson · Answer

undefined To solve the equation sqrt(x+5) + 1 = x, we can follow these steps:

1. Start by isolating the square root term on one side of the equation. Subtract 1 from both sides:
   sqrt(x+5) = x - 1

2. Square both sides of the equation to eliminate the square root:
   (sqrt(x+5))^2 = (x - 1)^2
   x + 5 = x^2 - 2x + 1

3. Rearrange the equation to form a quadratic equation:
   x^2 - 3x - 4 = 0

4. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we can factor the equation:
   (x - 4)(x + 1) = 0

5. Set each factor equal to zero and solve for x:
   x - 4 = 0  or  x + 1 = 0

x = 4  or  x = -1

Therefore, the solutions to the equation sqrt(x+5) + 1 = x are x = 4 and x = -1.