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Simplifying 0 = -16t2 + 90t + 1000 Reorder the terms: 0 = 1000 + 90t + -16t2 Solving 0 = 1000 + 90t + -16t2 Solving for variable 't'. Combine like terms: 0 + -1000 = -1000 -1000 + -90t + 16t2 = 1000 + 90t + -16t2 + -1000 + -90t + 16t2 Reorder the terms: -1000 + -90t + 16t2 = 1000 + -1000 + 90t + -90t + -16t2 + 16t2 Combine like terms: 1000 + -1000 = 0 -1000 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 -1000 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 -1000 + -90t + 16t2 = 0 + -16t2 + 16t2 -1000 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1000 + -90t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-500 + -45t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-500 + -45t + 8t2)' equal to zero and attempt to solve: Simplifying -500 + -45t + 8t2 = 0 Solving -500 + -45t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -62.5 + -5.625t + t2 = 0 Move the constant term to the right: Add '62.5' to each side of the equation. -62.5 + -5.625t + 62.5 + t2 = 0 + 62.5 Reorder the terms: -62.5 + 62.5 + -5.625t + t2 = 0 + 62.5 Combine like terms: -62.5 + 62.5 = 0.0 0.0 + -5.625t + t2 = 0 + 62.5 -5.625t + t2 = 0 + 62.5 Combine like terms: 0 + 62.5 = 62.5 -5.625t + t2 = 62.5 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 = 62.5 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = 62.5 + 7.91015625 Combine like terms: 62.5 + 7.91015625 = 70.41015625 7.91015625 + -5.625t + t2 = 70.41015625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 70.41015625 Calculate the square root of the right side: 8.391075989 Break this problem into two subproblems by setting (t + -2.8125) equal to 8.391075989 and -8.391075989.Subproblem 1
t + -2.8125 = 8.391075989 Simplifying t + -2.8125 = 8.391075989 Reorder the terms: -2.8125 + t = 8.391075989 Solving -2.8125 + t = 8.391075989 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 = 8.391075989 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 8.391075989 + 2.8125 t = 8.391075989 + 2.8125 Combine like terms: 8.391075989 + 2.8125 = 11.203575989 t = 11.203575989 Simplifying t = 11.203575989Subproblem 2
t + -2.8125 = -8.391075989 Simplifying t + -2.8125 = -8.391075989 Reorder the terms: -2.8125 + t = -8.391075989 Solving -2.8125 + t = -8.391075989 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 = -8.391075989 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -8.391075989 + 2.8125 t = -8.391075989 + 2.8125 Combine like terms: -8.391075989 + 2.8125 = -5.578575989 t = -5.578575989 Simplifying t = -5.578575989Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.203575989, -5.578575989}Solution
t = {11.203575989, -5.578575989}
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