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Simplifying 1000t = -6t2 + 15000 Reorder the terms: 1000t = 15000 + -6t2 Solving 1000t = 15000 + -6t2 Solving for variable 't'. Reorder the terms: -15000 + 1000t + 6t2 = 15000 + -6t2 + -15000 + 6t2 Reorder the terms: -15000 + 1000t + 6t2 = 15000 + -15000 + -6t2 + 6t2 Combine like terms: 15000 + -15000 = 0 -15000 + 1000t + 6t2 = 0 + -6t2 + 6t2 -15000 + 1000t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 -15000 + 1000t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7500 + 500t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7500 + 500t + 3t2)' equal to zero and attempt to solve: Simplifying -7500 + 500t + 3t2 = 0 Solving -7500 + 500t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2500 + 166.6666667t + t2 = 0 Move the constant term to the right: Add '2500' to each side of the equation. -2500 + 166.6666667t + 2500 + t2 = 0 + 2500 Reorder the terms: -2500 + 2500 + 166.6666667t + t2 = 0 + 2500 Combine like terms: -2500 + 2500 = 0 0 + 166.6666667t + t2 = 0 + 2500 166.6666667t + t2 = 0 + 2500 Combine like terms: 0 + 2500 = 2500 166.6666667t + t2 = 2500 The t term is 166.6666667t. Take half its coefficient (83.33333335). Square it (6944.444447) and add it to both sides. Add '6944.444447' to each side of the equation. 166.6666667t + 6944.444447 + t2 = 2500 + 6944.444447 Reorder the terms: 6944.444447 + 166.6666667t + t2 = 2500 + 6944.444447 Combine like terms: 2500 + 6944.444447 = 9444.444447 6944.444447 + 166.6666667t + t2 = 9444.444447 Factor a perfect square on the left side: (t + 83.33333335)(t + 83.33333335) = 9444.444447 Calculate the square root of the right side: 97.182531594 Break this problem into two subproblems by setting (t + 83.33333335) equal to 97.182531594 and -97.182531594.Subproblem 1
t + 83.33333335 = 97.182531594 Simplifying t + 83.33333335 = 97.182531594 Reorder the terms: 83.33333335 + t = 97.182531594 Solving 83.33333335 + t = 97.182531594 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-83.33333335' to each side of the equation. 83.33333335 + -83.33333335 + t = 97.182531594 + -83.33333335 Combine like terms: 83.33333335 + -83.33333335 = 0.00000000 0.00000000 + t = 97.182531594 + -83.33333335 t = 97.182531594 + -83.33333335 Combine like terms: 97.182531594 + -83.33333335 = 13.849198244 t = 13.849198244 Simplifying t = 13.849198244Subproblem 2
t + 83.33333335 = -97.182531594 Simplifying t + 83.33333335 = -97.182531594 Reorder the terms: 83.33333335 + t = -97.182531594 Solving 83.33333335 + t = -97.182531594 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-83.33333335' to each side of the equation. 83.33333335 + -83.33333335 + t = -97.182531594 + -83.33333335 Combine like terms: 83.33333335 + -83.33333335 = 0.00000000 0.00000000 + t = -97.182531594 + -83.33333335 t = -97.182531594 + -83.33333335 Combine like terms: -97.182531594 + -83.33333335 = -180.515864944 t = -180.515864944 Simplifying t = -180.515864944Solution
The solution to the problem is based on the solutions from the subproblems. t = {13.849198244, -180.515864944}Solution
t = {13.849198244, -180.515864944}
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