Solve the quadratic equation. Check the answers. \[ x^{2}+3 x-5=5 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution(s) is/are \( x= \) (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) B. There are no real solutions.

First, let's solve the quadratic equation \( x^{2} + 3x - 5 = 5 \) by first moving the terms around: \[ x^{2} + 3x - 10 = 0 \] Now we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = -10 \). Calculating the discriminant: \[ b^2 - 4ac = 3^2 - 4(1)(-10) = 9 + 40 = 49 \] This is positive, so we have two real solutions: \[ x = \frac{-3 \pm \sqrt{49}}{2(1)} \] \[ x = \frac{-3 \pm 7}{2} \] Now compute the two solutions: 1. \( x = \frac{4}{2} = 2 \) 2. \( x = \frac{-10}{2} = -5 \) Thus, the solutions are \( x = 2, -5 \). A. The solution(s) is/are \( x = 2, -5 \). --- Did you know that quadratic equations have been studied for over 2000 years? The ancient Babylonian mathematicians were already solving equations similar to these using geometric methods long before the advent of algebra as we know it today. Their techniques often involved cutting shapes into pieces to find solutions, a fascinating precursor to our modern-day algorithms! In the world of architecture and engineering, quadratic equations play a vital role. For instance, when designing arches, bridges, or parabolas in buildings, understanding the properties of these curves ensures stability and aesthetic appeal. So, the next time you marvel at a grand structure, remember that mathematics is often hidden behind its beauty!