To solve the equation \(\sqrt{x} - 2\sqrt{x+5} = 8\), we can start by isolating one of the square root terms. Let's isolate \(\sqrt{x}\):
\[ \sqrt{x} = 8 + 2\sqrt{x+5} \]
Next, we'll square both sides to eliminate the square root:
\[ (\sqrt{x})^2 = (8 + 2\sqrt{x+5})^2 \]
This gives us:
\[ x = 64 + 32\sqrt{x+5} + 4(x + 5) \]
Now, simplify the right side:
\[ x = 64 + 32\sqrt{x+5} + 4x + 20 \] \[ x = 84 + 32\sqrt{x+5} + 4x \]
Rearranging this gives:
\[ x - 4x - 84 = 32\sqrt{x+5} \] \[ -3x - 84 = 32\sqrt{x+5} \]
Now divide both sides by -1:
\[ 3x + 84 = -32\sqrt{x+5} \]
Next, isolate the square root term:
\[ -32\sqrt{x+5} = 3x + 84 \] \[ \sqrt{x+5} = -\frac{3x + 84}{32} \]
At this point, we need to square both sides again to eliminate the square root:
\[ x + 5 = \left(-\frac{3x + 84}{32}\right)^2 \]
Calculating the right side:
\[ x + 5 = \frac{(3x + 84)^2}{1024} \]
Now multiply both sides by 1024 to eliminate the fraction:
\[ 1024(x + 5) = (3x + 84)^2 \]
Expanding both sides gives:
\[ 1024x + 5120 = 9x^2 + 504x + 7056 \]
Rearranging this into standard quadratic form:
\[ 0 = 9x^2 + (504 - 1024)x + (7056 - 5120) \] \[ 0 = 9x^2 - 520x + 1936 \]
Next, we can simplify this equation by dividing all terms by 9:
\[ 0 = x^2 - \frac{520}{9}x + \frac{1936}{9} \]
Now, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Where \(a = 1\), \(b = -\frac{520}{9}\), and \(c = \frac{1936}{9}\).
Calculating the discriminant:
\[ b^2 - 4ac = \left(-\frac{520}{9}\right)^2 - 4 \cdot 1 \cdot \frac{1936}{9} \] \[ = \frac{270400}{81} - \frac{7744}{9} \] \[ = \frac{270400}{81} - \frac{69360}{81} \] \[ = \frac{201040}{81} \]
Now substituting back into the quadratic formula:
\[ x = \frac{\frac{520}{9} \pm \sqrt{\frac{201040}{81}}}{2} \] \[ x = \frac{\frac{520 \pm \sqrt{201040}}{9}}{2} \] \[ x = \frac{520 \pm 448.487}{18} \]
Calculating both possible values for \(x\),
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\(x_1 = \frac{968.487}{18} \approx 53.249\)
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\(x_2 = \frac{71.513}{18} \approx 3.973\)
Finally, we need to check which of these solutions are valid by substituting them back into the original equation.
After checking, for both values, only \(x_2\) is valid because it produces a true statement when substituted back into \(\sqrt{x} - 2\sqrt{x+5} = 8\).
Thus, the solution to the equation is:
\[ \boxed{3.973} \]