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Simplifying 0 = -16t2 + 90t + 10000 Reorder the terms: 0 = 10000 + 90t + -16t2 Solving 0 = 10000 + 90t + -16t2 Solving for variable 't'. Combine like terms: 0 + -10000 = -10000 -10000 + -90t + 16t2 = 10000 + 90t + -16t2 + -10000 + -90t + 16t2 Reorder the terms: -10000 + -90t + 16t2 = 10000 + -10000 + 90t + -90t + -16t2 + 16t2 Combine like terms: 10000 + -10000 = 0 -10000 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 -10000 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 -10000 + -90t + 16t2 = 0 + -16t2 + 16t2 -10000 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -10000 + -90t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5000 + -45t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5000 + -45t + 8t2)' equal to zero and attempt to solve: Simplifying -5000 + -45t + 8t2 = 0 Solving -5000 + -45t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -625 + -5.625t + t2 = 0 Move the constant term to the right: Add '625' to each side of the equation. -625 + -5.625t + 625 + t2 = 0 + 625 Reorder the terms: -625 + 625 + -5.625t + t2 = 0 + 625 Combine like terms: -625 + 625 = 0 0 + -5.625t + t2 = 0 + 625 -5.625t + t2 = 0 + 625 Combine like terms: 0 + 625 = 625 -5.625t + t2 = 625 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 = 625 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = 625 + 7.91015625 Combine like terms: 625 + 7.91015625 = 632.91015625 7.91015625 + -5.625t + t2 = 632.91015625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 632.91015625 Calculate the square root of the right side: 25.157705703 Break this problem into two subproblems by setting (t + -2.8125) equal to 25.157705703 and -25.157705703.Subproblem 1
t + -2.8125 = 25.157705703 Simplifying t + -2.8125 = 25.157705703 Reorder the terms: -2.8125 + t = 25.157705703 Solving -2.8125 + t = 25.157705703 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 = 25.157705703 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 25.157705703 + 2.8125 t = 25.157705703 + 2.8125 Combine like terms: 25.157705703 + 2.8125 = 27.970205703 t = 27.970205703 Simplifying t = 27.970205703Subproblem 2
t + -2.8125 = -25.157705703 Simplifying t + -2.8125 = -25.157705703 Reorder the terms: -2.8125 + t = -25.157705703 Solving -2.8125 + t = -25.157705703 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 = -25.157705703 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -25.157705703 + 2.8125 t = -25.157705703 + 2.8125 Combine like terms: -25.157705703 + 2.8125 = -22.345205703 t = -22.345205703 Simplifying t = -22.345205703Solution
The solution to the problem is based on the solutions from the subproblems. t = {27.970205703, -22.345205703}Solution
t = {27.970205703, -22.345205703}
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