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Simplifying 90 = -16t2 + 90t + 2 Reorder the terms: 90 = 2 + 90t + -16t2 Solving 90 = 2 + 90t + -16t2 Solving for variable 't'. Combine like terms: 90 + -2 = 88 88 + -90t + 16t2 = 2 + 90t + -16t2 + -2 + -90t + 16t2 Reorder the terms: 88 + -90t + 16t2 = 2 + -2 + 90t + -90t + -16t2 + 16t2 Combine like terms: 2 + -2 = 0 88 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 88 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 88 + -90t + 16t2 = 0 + -16t2 + 16t2 88 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 88 + -90t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(44 + -45t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(44 + -45t + 8t2)' equal to zero and attempt to solve: Simplifying 44 + -45t + 8t2 = 0 Solving 44 + -45t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 5.5 + -5.625t + t2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + -5.625t + -5.5 + t2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + -5.625t + t2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + -5.625t + t2 = 0 + -5.5 -5.625t + t2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 -5.625t + t2 = -5.5 The t term is -5.625t. Take half its coefficient (-2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. -5.625t + 7.91015625 + t2 = -5.5 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = -5.5 + 7.91015625 Combine like terms: -5.5 + 7.91015625 = 2.41015625 7.91015625 + -5.625t + t2 = 2.41015625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 2.41015625 Calculate the square root of the right side: 1.552467794 Break this problem into two subproblems by setting (t + -2.8125) equal to 1.552467794 and -1.552467794.Subproblem 1
t + -2.8125 = 1.552467794 Simplifying t + -2.8125 = 1.552467794 Reorder the terms: -2.8125 + t = 1.552467794 Solving -2.8125 + t = 1.552467794 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = 1.552467794 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 1.552467794 + 2.8125 t = 1.552467794 + 2.8125 Combine like terms: 1.552467794 + 2.8125 = 4.364967794 t = 4.364967794 Simplifying t = 4.364967794Subproblem 2
t + -2.8125 = -1.552467794 Simplifying t + -2.8125 = -1.552467794 Reorder the terms: -2.8125 + t = -1.552467794 Solving -2.8125 + t = -1.552467794 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = -1.552467794 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -1.552467794 + 2.8125 t = -1.552467794 + 2.8125 Combine like terms: -1.552467794 + 2.8125 = 1.260032206 t = 1.260032206 Simplifying t = 1.260032206Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.364967794, 1.260032206}Solution
t = {4.364967794, 1.260032206}
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