X-√x+1=5

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

x - √x + 1 = 5

Subtracting 1 from both sides:

x - √x = 4

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

(x - √x)^2 = 4^2

Expanding the left side of the equation:

x^2 - 2√x * x + (√x)^2 = 16

x^2 - 2x√x + x = 16

Combining like terms:

x^2 - 2x√x + x - 16 = 0

Rearranging the terms:

x^2 + x - 2x√x - 16 = 0

Let's now isolate the square root term by moving all other terms to the other side:

2x√x = x^2 + x - 16

Now, let's square both sides again to eliminate the square root:

(2x√x)^2 = (x^2 + x - 16)^2

4x^2 * x = (x^2 + x - 16)(x^2 + x - 16)

4x^3 = (x^2 + x - 16)(x^2 + x - 16)

Expanding the right side of the equation:

4x^3 = x^4 + x^3 - 16x^2 + x^3 + x^2 - 16x - 16x^2 - 16x + 256

Simplifying:

4x^3 = x^4 + 2x^3 - 32x^2 - 32x + 256

Now, let's rearrange the terms to form a quadratic equation:

x^4 - 2x^3 - 32x^2 - 32x + 256 - 4x^3 = 0

x^4 - 6x^3 - 32x^2 - 32x + 256 = 0

At this point, it's difficult to solve this equation algebraically. You may want to use numerical methods (such as factoring, long division, or graphing) or a calculator to approximate the possible solutions.