Final answer:
The problem is about solving a differential equation and a quadratic equation in Mathematics for High School level. The differential equation should be approached with integrating factors, whereas the quadratic can be solved using the quadratic formula with the appropriate coefficients.
Step-by-step explanation:
The question provided falls under the subject of Mathematics and is most likely targeted at High School level students, discussing how to find the general solution to a first-order linear differential equation and then separately how to solve a quadratic equation using the quadratic formula.
General Solution to a First-order Linear Differential Equation
The differential equation provided is t dy/dt - 2ty = t² - t. This can be rearranged to a standard first order linear differential equation by dividing through by t (assuming t ≠ 0), leading to dy/dt - 2y = t - 1. This equation can then be solved using standard techniques for first-order linear differential equations, such as finding an integrating factor.
Solving a Quadratic Equation
To solve the quadratic equation t² + 10t - 2000 = 0 or t² + 10t - 200 = 0 using the quadratic formula, the formula t= (-b ± √(b² - 4ac))/(2a) can be applied, where a = 1, b = 10, and c = -2000 or c = -200 corresponding to the different typos in the question. The roots t of the equation are then found by substituting these values into the formula.