To solve the equation x - √(x+1) = -5/4, we can start by isolating the square root term.

x - √(x+1) = -5/4

Add √(x+1) to both sides:

x = -5/4 + √(x+1)

Next, let's isolate the square root term on the right side:

x + 5/4 = √(x+1)

Now, square both sides of the equation to eliminate the square root:

(x + 5/4)^2 = (√(x+1))^2

(x + 5/4)^2 = x + 1

Expand the left side:

(x^2 + 5/2x + 25/16) = x + 1

Now, let's simplify the equation:

x^2 + 5/2x + 25/16 = x + 1

Multiply every term by 16 to get rid of the fractions:

16x^2 + 40x + 25 = 16x + 16

Rearrange the equation:

16x^2 + 40x - 16x + 25 - 16 = 0

Combine like terms:

16x^2 + 24x + 9 = 0

Now, we can solve this quadratic equation. However, it does not factor nicely, so we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 16, b = 24, and c = 9. Plugging these values into the quadratic formula:

x = (-24 ± √(24^2 - 4(16)(9))) / (2(16))

x = (-24 ± √(576 - 576)) / 32

x = (-24 ± √0) / 32

Since the discriminant (√(b^2 - 4ac)) is 0, there is only one solution:

x = -24/32

Simplifying the fraction:

x = -3/4

Therefore, the solution to the equation x - √(x+1) = -5/4 is x = -3/4.