Final answer:
To solve the quadratic equation for t, we use the quadratic formula with the values a = 1, b = 10, and c = -2000, yielding two solutions for t: 40 and -50.
Step-by-step explanation:
To solve the given quadratic equation for t, we can apply the quadratic formula. First, we must confirm that the equation is in the standard form of at² + bt + c = 0. The provided equations are variations on this form:
- t² + 10t - 2000 = 0
- t² + 10t - 200 = 0
We'll take the first equation as an example. To use the quadratic formula, t = [-b ± √(b² - 4ac)] / (2a), where a = 1, b = 10, and c = -2000.
Plugging the values into the quadratic formula, we get:
t = [-10 ± √(10² - 4(1)(-2000))] / (2· 1)
t = [-10 ± √(100 + 8000)] / 2
t = [-10 ± √(8100)] / 2
t = [-10 ± 90] / 2
Therefore, we have two solutions for t:
- t = (-10 + 90) / 2 = 40
- t = (-10 - 90) / 2 = -50
These two values are the solutions for t in the equation t² + 10t - 2000 = 0.