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Simplifying 100 = 6.2t2 + 11t + 32 Reorder the terms: 100 = 32 + 11t + 6.2t2 Solving 100 = 32 + 11t + 6.2t2 Solving for variable 't'. Combine like terms: 100 + -32 = 68 68 + -11t + -6.2t2 = 32 + 11t + 6.2t2 + -32 + -11t + -6.2t2 Reorder the terms: 68 + -11t + -6.2t2 = 32 + -32 + 11t + -11t + 6.2t2 + -6.2t2 Combine like terms: 32 + -32 = 0 68 + -11t + -6.2t2 = 0 + 11t + -11t + 6.2t2 + -6.2t2 68 + -11t + -6.2t2 = 11t + -11t + 6.2t2 + -6.2t2 Combine like terms: 11t + -11t = 0 68 + -11t + -6.2t2 = 0 + 6.2t2 + -6.2t2 68 + -11t + -6.2t2 = 6.2t2 + -6.2t2 Combine like terms: 6.2t2 + -6.2t2 = 0.0 68 + -11t + -6.2t2 = 0.0 Begin completing the square. Divide all terms by -6.2 the coefficient of the squared term: Divide each side by '-6.2'. -10.96774194 + 1.774193548t + t2 = 0 Move the constant term to the right: Add '10.96774194' to each side of the equation. -10.96774194 + 1.774193548t + 10.96774194 + t2 = 0 + 10.96774194 Reorder the terms: -10.96774194 + 10.96774194 + 1.774193548t + t2 = 0 + 10.96774194 Combine like terms: -10.96774194 + 10.96774194 = 0.00000000 0.00000000 + 1.774193548t + t2 = 0 + 10.96774194 1.774193548t + t2 = 0 + 10.96774194 Combine like terms: 0 + 10.96774194 = 10.96774194 1.774193548t + t2 = 10.96774194 The t term is 1.774193548t. Take half its coefficient (0.887096774). Square it (0.7869406864) and add it to both sides. Add '0.7869406864' to each side of the equation. 1.774193548t + 0.7869406864 + t2 = 10.96774194 + 0.7869406864 Reorder the terms: 0.7869406864 + 1.774193548t + t2 = 10.96774194 + 0.7869406864 Combine like terms: 10.96774194 + 0.7869406864 = 11.7546826264 0.7869406864 + 1.774193548t + t2 = 11.7546826264 Factor a perfect square on the left side: (t + 0.887096774)(t + 0.887096774) = 11.7546826264 Calculate the square root of the right side: 3.428510263 Break this problem into two subproblems by setting (t + 0.887096774) equal to 3.428510263 and -3.428510263.Subproblem 1
t + 0.887096774 = 3.428510263 Simplifying t + 0.887096774 = 3.428510263 Reorder the terms: 0.887096774 + t = 3.428510263 Solving 0.887096774 + t = 3.428510263 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.887096774' to each side of the equation. 0.887096774 + -0.887096774 + t = 3.428510263 + -0.887096774 Combine like terms: 0.887096774 + -0.887096774 = 0.000000000 0.000000000 + t = 3.428510263 + -0.887096774 t = 3.428510263 + -0.887096774 Combine like terms: 3.428510263 + -0.887096774 = 2.541413489 t = 2.541413489 Simplifying t = 2.541413489Subproblem 2
t + 0.887096774 = -3.428510263 Simplifying t + 0.887096774 = -3.428510263 Reorder the terms: 0.887096774 + t = -3.428510263 Solving 0.887096774 + t = -3.428510263 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.887096774' to each side of the equation. 0.887096774 + -0.887096774 + t = -3.428510263 + -0.887096774 Combine like terms: 0.887096774 + -0.887096774 = 0.000000000 0.000000000 + t = -3.428510263 + -0.887096774 t = -3.428510263 + -0.887096774 Combine like terms: -3.428510263 + -0.887096774 = -4.315607037 t = -4.315607037 Simplifying t = -4.315607037Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.541413489, -4.315607037}
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