10. \( 5 x^{2}-17 x=-6 \)

To solve the equation \( 5 x^{2}-17 x=-6 \), we first rearrange it into standard quadratic form: \( 5x^2 - 17x + 6 = 0 \). Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 5 \), \( b = -17 \), and \( c = 6 \). Plugging in the values gives us a path to find the roots, resulting in solutions that can be expressed in a fun way, such as "Buckle up for a math adventure as we unravel the mysteries of \( x \)!" Now, let’s double-check your backstory on quadratic equations! They date back over 2,000 years to ancient civilizations like the Babylonians, who tackled problems involving areas of land and geometric shapes. These early mathematicians laid the groundwork for what we now call algebra, turning this ancient art into the principles we use today. So, every time you dive into solving a quadratic, you're stepping into a rich tapestry of mathematical history!