Let's solve the equation \(x^2 - x = 870\) using the **Babylonian (or Heron's) method**, which is an ancient method for finding square roots. ### **Step 1: Rearrange and Estimate** First, rewrite the equation: \[ x^2 - x = 870 \implies x^2 - x - 870 = 0 \] Assume \(x\) is large, so \(x^2 \approx 870\). Let's estimate \(x\): \[ x^2 \approx 870 \implies x \approx \sqrt{870} \approx 29.5 \] **Explanation:** The Babylonian method starts with an initial guess for the square root (here, of 870). The closest integer is 30. --- ### **Step 2: Refine Estimate Using Babylonian Iteration** Let’s use the Babylonian formula to refine our guess. For the equation \(x^2 - x = 870\), rewrite as: \[ x^2 = 870 + x \] So, \[ x = \sqrt{870 + x} \] Let’s use \(x_0 = 30\) as our initial guess. **Babylonian iteration:** \[ x_{n+1} = \frac{1}{2} \left(x_n + \frac{870 + x_n}{x_n}\right) \] **Calculate \(x_1\):** \[ x_1 = \frac{1}{2}\left(30 + \frac{870 + 30}{30}\right) = \frac{1}{2}\left(30 + \frac{900}{30}\right) = \frac{1}{2}(30 + 30) = 30 \] So the first iteration doesn’t change much. Try with \(x_0 = 29.5\): \[ x_1 = \frac{1}{2}\left(29.5 + \frac{870 + 29.5}{29.5}\right) = \frac{1}{2}\left(29.5 + \frac{899.5}{29.5}\right) \approx \frac{1}{2}(29.5 + 30.5) = \frac{1}{2}(60) = 30 \] So, \(x \approx 30\) is a good approximation. **Explanation:** Plugging \(x=30\) back into the original equation: \[ 30^2 - 30 = 900 - 30 = 870 \] It fits exactly. --- ### **Final Answer** The solution to \(x^2 - x = 870\) using the Babylonian approach is: \[ \boxed{x = 30} \] If you consider the quadratic equation \(x^2 - x - 870 = 0\), the other root (using the quadratic formula) would be negative and not relevant for Babylonian square-root style iteration.