1. \( 15 x^{2}-x-20=3 x^{2} \)

To solve the equation, start by rearranging it to one side: \( 15x^2 - x - 20 - 3x^2 = 0 \), which simplifies to \( 12x^2 - x - 20 = 0 \). Now, you can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 12, b = -1, c = -20 \). This gives you \( x = \frac{1 \pm \sqrt{1 + 960}}{24} = \frac{1 \pm \sqrt{961}}{24} \). Since \( \sqrt{961} = 31 \), the solutions are \( x = \frac{32}{24} = \frac{4}{3} \) and \( x = \frac{-30}{24} = -\frac{5}{4} \). The equation provides a classic example of a quadratic equation where rearranging is key to simplifying the problem. Quadratics show up in various contexts, from physics to finance, making it essential for students and professionals alike to grasp these concepts. Embracing the beauty of algebra not only helps with schoolwork, but it also sharpens critical thinking skills for real-life challenges!