Final answer:
To solve the quadratic equation t² + 10t - 200 = 0, the quadratic formula is used, yielding two solutions for t: 10 and -20. Considering physical context, t = 10 seconds is the relevant solution.
Step-by-step explanation:
The student is asked to find a general solution to a given differential equation, which typically involves integration or applying known methods such as separation of variables, integrating factors, characteristic equations, etc. The student is also given a quadratic equation, t² + 10t - 200 or t² + 10t - 2000, to solve for the variable t. To solve for t, the quadratic formula is used, which requires setting the equation to zero and identifying coefficients a, b, and c.
First, the quadratic equation needs to be in standard form:
t² + 10t - 200 = 0
Here the coefficients are a = 1, b = 10, and c = -200. The quadratic formula t = (-b ± √(b²-4ac)) / (2a) will give the values of t. Applying the formula:
t = (-10 ± √(10²-4×1×(-200))) / (2×1)
t = (-10 ± √(100 + 800)) / 2
t = (-10 ± √(900)) / 2
t = (-10 ± 30) / 2
which provides two possible solutions for t:
- t = (30 - 10) / 2 = 10
- t = (-30 - 10) / 2 = -20
Since negative time is usually not physical in these contexts, t = 10 seconds is likely the relevant solution.