1. \( x^{2}=6 x+91 \)

To solve the equation \( x^{2} = 6x + 91 \), start by rearranging it into standard quadratic form: \( x^{2} - 6x - 91 = 0 \). Now, you can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). Here, \( a = 1, b = -6, \) and \( c = -91 \). This will give you two potential solutions for \( x \). After calculating the discriminant \( b^{2} - 4ac \), you’ll find it's a positive number, indicating that both solutions are real and distinct. Plug those values into the quadratic formula, and voilà—you'll have your solutions for \( x \)!