Solve for all solutions of x : \[ x^{2}-10 x+30=0 \] \( x=5 \pm \sqrt{5} \) \( x=5 \pm \sqrt{55} \) \( x=5 \pm \sqrt{30} \) \( x=5 \pm i \sqrt{5} \)

To solve the quadratic equation \(x^{2}-10x+30=0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we first identify \(a = 1\), \(b = -10\), and \(c = 30\). Calculating the discriminant: \[ b^2 - 4ac = (-10)^2 - 4 \cdot 1 \cdot 30 = 100 - 120 = -20 \] Since the discriminant is negative, this indicates that there are two complex solutions. Now we calculate: \[ x = \frac{-(-10) \pm \sqrt{-20}}{2 \cdot 1} = \frac{10 \pm \sqrt{-20}}{2} = \frac{10 \pm 2i\sqrt{5}}{2} = 5 \pm i\sqrt{5} \] Thus, the solutions are: \[ x = 5 \pm i\sqrt{5} \] If you’ve ever wondered how the quadratic formula arose, it actually comes from completing the square! By rewriting \(x^{2} - 10x + 30\) into a perfect square form, you can derive the same solution beautifully without diving straight into the formula. Now, let’s talk about complex solutions in real-world applications! These solutions often come into play in engineering, particularly in analyzing AC circuits where they model impedance. Complex numbers allow engineers to seamlessly handle phase shifts and amplitude in sinusoidal functions – who knew math could be so electrifying?