Step 1 :Given the quadratic equation: \(x^2 - x + 3\sqrt{3} - 5 = 0\), we can identify the coefficients as \(a = 1\), \(b = -1\), and \(c = 3\sqrt{3} - 5\).
Step 2 :Using the Bhaskara formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we first calculate the discriminant: \(b^2 - 4ac = (-1)^2 - 4(1)(3\sqrt{3} - 5)\).
Step 3 :Calculating the discriminant, we get: \(b^2 - 4ac \approx 0.215\).
Step 4 :Now, we can find the roots using the Bhaskara formula: \(x_1 = \frac{-(-1) + \sqrt{0.215}}{2(1)}\) and \(x_2 = \frac{-(-1) - \sqrt{0.215}}{2(1)}\).
Step 5 :Calculating the roots, we get: \(x_1 \approx 0.732\) and \(x_2 \approx 0.268\).
Step 6 :\(\boxed{x_1 \approx 0.732, x_2 \approx 0.268}\)