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Simplifying 0 = 82 + 13.5t + -4.9t2 Solving 0 = 82 + 13.5t + -4.9t2 Solving for variable 't'. Combine like terms: 0 + -82 = -82 -82 + -13.5t + 4.9t2 = 82 + 13.5t + -4.9t2 + -82 + -13.5t + 4.9t2 Reorder the terms: -82 + -13.5t + 4.9t2 = 82 + -82 + 13.5t + -13.5t + -4.9t2 + 4.9t2 Combine like terms: 82 + -82 = 0 -82 + -13.5t + 4.9t2 = 0 + 13.5t + -13.5t + -4.9t2 + 4.9t2 -82 + -13.5t + 4.9t2 = 13.5t + -13.5t + -4.9t2 + 4.9t2 Combine like terms: 13.5t + -13.5t = 0.0 -82 + -13.5t + 4.9t2 = 0.0 + -4.9t2 + 4.9t2 -82 + -13.5t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 -82 + -13.5t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. -16.73469388 + -2.755102041t + t2 = 0 Move the constant term to the right: Add '16.73469388' to each side of the equation. -16.73469388 + -2.755102041t + 16.73469388 + t2 = 0 + 16.73469388 Reorder the terms: -16.73469388 + 16.73469388 + -2.755102041t + t2 = 0 + 16.73469388 Combine like terms: -16.73469388 + 16.73469388 = 0.00000000 0.00000000 + -2.755102041t + t2 = 0 + 16.73469388 -2.755102041t + t2 = 0 + 16.73469388 Combine like terms: 0 + 16.73469388 = 16.73469388 -2.755102041t + t2 = 16.73469388 The t term is -2.755102041t. Take half its coefficient (-1.377551021). Square it (1.897646815) and add it to both sides. Add '1.897646815' to each side of the equation. -2.755102041t + 1.897646815 + t2 = 16.73469388 + 1.897646815 Reorder the terms: 1.897646815 + -2.755102041t + t2 = 16.73469388 + 1.897646815 Combine like terms: 16.73469388 + 1.897646815 = 18.632340695 1.897646815 + -2.755102041t + t2 = 18.632340695 Factor a perfect square on the left side: (t + -1.377551021)(t + -1.377551021) = 18.632340695 Calculate the square root of the right side: 4.316519512 Break this problem into two subproblems by setting (t + -1.377551021) equal to 4.316519512 and -4.316519512.Subproblem 1
t + -1.377551021 = 4.316519512 Simplifying t + -1.377551021 = 4.316519512 Reorder the terms: -1.377551021 + t = 4.316519512 Solving -1.377551021 + t = 4.316519512 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.377551021' to each side of the equation. -1.377551021 + 1.377551021 + t = 4.316519512 + 1.377551021 Combine like terms: -1.377551021 + 1.377551021 = 0.000000000 0.000000000 + t = 4.316519512 + 1.377551021 t = 4.316519512 + 1.377551021 Combine like terms: 4.316519512 + 1.377551021 = 5.694070533 t = 5.694070533 Simplifying t = 5.694070533Subproblem 2
t + -1.377551021 = -4.316519512 Simplifying t + -1.377551021 = -4.316519512 Reorder the terms: -1.377551021 + t = -4.316519512 Solving -1.377551021 + t = -4.316519512 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.377551021' to each side of the equation. -1.377551021 + 1.377551021 + t = -4.316519512 + 1.377551021 Combine like terms: -1.377551021 + 1.377551021 = 0.000000000 0.000000000 + t = -4.316519512 + 1.377551021 t = -4.316519512 + 1.377551021 Combine like terms: -4.316519512 + 1.377551021 = -2.938968491 t = -2.938968491 Simplifying t = -2.938968491Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.694070533, -2.938968491}
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