Final answer:
The correct answer to the equation (500 = 4(7 - 2)^2) is not provided in the listed options. However, if the student is looking for the solution to t² + 10t - 200 = 0 using the quadratic formula, then the solution is t = 10.
Step-by-step explanation:
The equation in question is (500 = 4(7 - 2)^2), which needs to be solved for the inner term. First, calculate the term within the parentheses, which is (7 - 2), resulting in 5. The equation then becomes (500 = 4 * 5^2). Now, 5^2 is 25, and multiplying this by 4 gives us 100. However, since we have 500 on the left side, this does not seem correct. Let's reconsider the approach.
If we have an equation in the form t² + 10t - 200, we can solve for t using the quadratic formula. The quadratic formula is typically written as t = (-b ± √(b² - 4ac))/(2a). In our case, a=1, b=10, and c=-200, resulting in the quadratic equation t² + 10t - 200 = 0.
By substituting these values into the quadratic formula, we get t = (-10 ± √(100 + 800))/2. Simplifying inside the square root gives us t = (-10 ± √900)/2. The square root of 900 is 30, so we have t = (-10 ± 30)/2, which gives us the two possible solutions: t = 20/2 or t = -40/2, yielding t = 10 or t = -20. However, the student's question only lists positive solutions, so the correct solution is t = 10.