(i) Start with the equation \(\log_{10}(x-4) = 2 - \log_{10} x\).

Using the law of logarithms for subtraction, \(\log_{10}(x-4) + \log_{10} x = 2\).

This can be rewritten as \(\log_{10}((x-4)x) = 2\).

Since \(\log_{10} 100 = 2\), we have \((x-4)x = 100\).

Expanding gives \(x^2 - 4x = 100\).

Rearranging, we obtain the quadratic equation \(x^2 - 4x - 100 = 0\).

(ii) Solve the quadratic equation \(x^2 - 4x - 100 = 0\).

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -100\).

Calculate the discriminant: \(b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-100) = 16 + 400 = 416\).

Thus, \(x = \frac{4 \pm \sqrt{416}}{2}\).

Calculate \(\sqrt{416} \approx 20.396\).

So, \(x = \frac{4 \pm 20.396}{2}\).

This gives solutions \(x = 12.198\) and \(x = -8.198\).

Since \(x\) must be positive, the solution is \(x = 12.2\) (to 3 significant figures).