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Determine t, the general term of the pattern, in the form t₁ = an² bn c given that t = -4n² 56n-180, determine the biggest numerical value for t₁.

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Final answer:

To solve the quadratic equation t² + 10t - 2000 = 0, apply the quadratic formula, identify a, b, and c from the equation, and then find the roots. Furthermore, for finding the maximum value of this function, determine the vertex of the quadratic parabola.

Step-by-step explanation:

The student is dealing with a quadratic equation in the form of t² + 10t - 2000 = 0 (according to one part of the question) or t² + 10t - 200 = 0 (according to another part). Assuming the correct equation is t² + 10t - 2000 = 0, we can use the quadratic formula to find the values of t:

Step-by-Step Explanation Using the Quadratic Formula:

  1. Rearrange the quadratic equation so that it is in the form at² + bt + c = 0, where a, b, and c are constants.
  2. Identify the coefficients: a = 1, b = 10, c = -2000.
  3. Apply the quadratic formula: t = -b ± √(b² - 4ac) / (2a).
  4. Substitute the identified coefficients into the formula and solve for t.
  5. Find the two possible solutions for t which will be the x-intercepts of the parabola.

To find the maximum value of t, we need to identify the vertex of the parabola, which is the peak point for a downwards opening parabola. This can be done using the formula -b / (2a) and evaluating the original quadratic equation at that point.

User Ghonima
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