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Simplifying 1000t = -6t2 + 1500 Reorder the terms: 1000t = 1500 + -6t2 Solving 1000t = 1500 + -6t2 Solving for variable 't'. Reorder the terms: -1500 + 1000t + 6t2 = 1500 + -6t2 + -1500 + 6t2 Reorder the terms: -1500 + 1000t + 6t2 = 1500 + -1500 + -6t2 + 6t2 Combine like terms: 1500 + -1500 = 0 -1500 + 1000t + 6t2 = 0 + -6t2 + 6t2 -1500 + 1000t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 -1500 + 1000t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-750 + 500t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-750 + 500t + 3t2)' equal to zero and attempt to solve: Simplifying -750 + 500t + 3t2 = 0 Solving -750 + 500t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -250 + 166.6666667t + t2 = 0 Move the constant term to the right: Add '250' to each side of the equation. -250 + 166.6666667t + 250 + t2 = 0 + 250 Reorder the terms: -250 + 250 + 166.6666667t + t2 = 0 + 250 Combine like terms: -250 + 250 = 0 0 + 166.6666667t + t2 = 0 + 250 166.6666667t + t2 = 0 + 250 Combine like terms: 0 + 250 = 250 166.6666667t + t2 = 250 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 = 250 + 6944.444447 Reorder the terms: 6944.444447 + 166.6666667t + t2 = 250 + 6944.444447 Combine like terms: 250 + 6944.444447 = 7194.444447 6944.444447 + 166.6666667t + t2 = 7194.444447 Factor a perfect square on the left side: (t + 83.33333335)(t + 83.33333335) = 7194.444447 Calculate the square root of the right side: 84.820071015 Break this problem into two subproblems by setting (t + 83.33333335) equal to 84.820071015 and -84.820071015.Subproblem 1
t + 83.33333335 = 84.820071015 Simplifying t + 83.33333335 = 84.820071015 Reorder the terms: 83.33333335 + t = 84.820071015 Solving 83.33333335 + t = 84.820071015 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 = 84.820071015 + -83.33333335 Combine like terms: 83.33333335 + -83.33333335 = 0.00000000 0.00000000 + t = 84.820071015 + -83.33333335 t = 84.820071015 + -83.33333335 Combine like terms: 84.820071015 + -83.33333335 = 1.486737665 t = 1.486737665 Simplifying t = 1.486737665Subproblem 2
t + 83.33333335 = -84.820071015 Simplifying t + 83.33333335 = -84.820071015 Reorder the terms: 83.33333335 + t = -84.820071015 Solving 83.33333335 + t = -84.820071015 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 = -84.820071015 + -83.33333335 Combine like terms: 83.33333335 + -83.33333335 = 0.00000000 0.00000000 + t = -84.820071015 + -83.33333335 t = -84.820071015 + -83.33333335 Combine like terms: -84.820071015 + -83.33333335 = -168.153404365 t = -168.153404365 Simplifying t = -168.153404365Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.486737665, -168.153404365}Solution
t = {1.486737665, -168.153404365}
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