1 Use the quadratic formula to find the real solutions, if any, of the equation. \[ x^{2}+2 x-6=0 \] Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is \( \{\square \). (Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. There are no real solutions.

To solve the equation \( x^{2}+2 x-6=0 \) using the quadratic formula, we identify \( a = 1 \), \( b = 2 \), and \( c = -6 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Calculating the discriminant, \( b^2 - 4ac \): \[ 2^2 - 4(1)(-6) = 4 + 24 = 28 \] Now substituting back into the formula: \[ x = \frac{-2 \pm \sqrt{28}}{2(1)} = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7} \] Thus, the solution set is: A. The solution set is \(\{-1 + \sqrt{7}, -1 - \sqrt{7}\}\). --- Quadratic equations have played a crucial role in the development of algebra from ancient times, with mathematicians such as Babylonian scholars using geometric methods to solve them over 4000 years ago! The quadratic formula itself was refined over centuries, leading to a robust tool we still use extensively in education today. Ever struggled with sign errors? A common mistake when using the quadratic formula is miscalculating the discriminant or mixing up the plus and minus signs. Always double-check your arithmetic and remember that the discriminant can tell you not only about the number of solutions but also their nature (real or complex), providing insight into the equation's behavior. Happy solving!