10. \( 8 x^{2}-2 x-10 \)

To solve the quadratic equation \( 8x^{2}-2x-10 = 0 \), you can first simplify it by dividing all terms by 2, which gives \( 4x^{2}-x-5 = 0 \). Next, you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4, b = -1, \) and \( c = -5 \). Plugging these values in helps you find the solutions for \( x \). Did you know that quadratic equations have been studied since ancient civilizations? The Babylonians were some of the first mathematicians to work with these types of equations around 2000 BCE, using geometric methods to find solutions. This legacy of math has paved the way for modern algebra and continues to influence various fields like physics and engineering to this day! Quadratic equations aren’t just theoretical; they pop up in real life quite often! For example, when physicists calculate projectile motion, they use quadratics to determine the object's path - think of a basketball shot or a cannonball's trajectory! So, understanding how to solve these equations can be invaluable not just in class but also in real-world applications like engineering, finance, and even video game design!