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Simplifying 100 = -16t2 + 90t + 60 Reorder the terms: 100 = 60 + 90t + -16t2 Solving 100 = 60 + 90t + -16t2 Solving for variable 't'. Combine like terms: 100 + -60 = 40 40 + -90t + 16t2 = 60 + 90t + -16t2 + -60 + -90t + 16t2 Reorder the terms: 40 + -90t + 16t2 = 60 + -60 + 90t + -90t + -16t2 + 16t2 Combine like terms: 60 + -60 = 0 40 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 40 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 40 + -90t + 16t2 = 0 + -16t2 + 16t2 40 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 40 + -90t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(20 + -45t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(20 + -45t + 8t2)' equal to zero and attempt to solve: Simplifying 20 + -45t + 8t2 = 0 Solving 20 + -45t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 2.5 + -5.625t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -5.625t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -5.625t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -5.625t + t2 = 0 + -2.5 -5.625t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -5.625t + t2 = -2.5 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 = -2.5 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = -2.5 + 7.91015625 Combine like terms: -2.5 + 7.91015625 = 5.41015625 7.91015625 + -5.625t + t2 = 5.41015625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 5.41015625 Calculate the square root of the right side: 2.325974258 Break this problem into two subproblems by setting (t + -2.8125) equal to 2.325974258 and -2.325974258.Subproblem 1
t + -2.8125 = 2.325974258 Simplifying t + -2.8125 = 2.325974258 Reorder the terms: -2.8125 + t = 2.325974258 Solving -2.8125 + t = 2.325974258 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 = 2.325974258 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 2.325974258 + 2.8125 t = 2.325974258 + 2.8125 Combine like terms: 2.325974258 + 2.8125 = 5.138474258 t = 5.138474258 Simplifying t = 5.138474258Subproblem 2
t + -2.8125 = -2.325974258 Simplifying t + -2.8125 = -2.325974258 Reorder the terms: -2.8125 + t = -2.325974258 Solving -2.8125 + t = -2.325974258 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 = -2.325974258 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -2.325974258 + 2.8125 t = -2.325974258 + 2.8125 Combine like terms: -2.325974258 + 2.8125 = 0.486525742 t = 0.486525742 Simplifying t = 0.486525742Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.138474258, 0.486525742}Solution
t = {5.138474258, 0.486525742}
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