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Simplifying (5t * t) + -1(15t) + -8400 = 0 Multiply t * t (5t2) + -1(15t) + -8400 = 0 Remove parenthesis around (15t) (5t2) + -1 * 15t + -8400 = 0 Multiply -1 * 15 (5t2) + -15t + -8400 = 0 Reorder the terms: -8400 + -15t + (5t2) = 0 Solving -8400 + -15t + (5t2) = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(-1680 + -3t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-1680 + -3t + t2)' equal to zero and attempt to solve: Simplifying -1680 + -3t + t2 = 0 Solving -1680 + -3t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '1680' to each side of the equation. -1680 + -3t + 1680 + t2 = 0 + 1680 Reorder the terms: -1680 + 1680 + -3t + t2 = 0 + 1680 Combine like terms: -1680 + 1680 = 0 0 + -3t + t2 = 0 + 1680 -3t + t2 = 0 + 1680 Combine like terms: 0 + 1680 = 1680 -3t + t2 = 1680 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 1680 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 1680 + 2.25 Combine like terms: 1680 + 2.25 = 1682.25 2.25 + -3t + t2 = 1682.25 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1682.25 Calculate the square root of the right side: 41.01524107 Break this problem into two subproblems by setting (t + -1.5) equal to 41.01524107 and -41.01524107.Subproblem 1
t + -1.5 = 41.01524107 Simplifying t + -1.5 = 41.01524107 Reorder the terms: -1.5 + t = 41.01524107 Solving -1.5 + t = 41.01524107 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 41.01524107 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 41.01524107 + 1.5 t = 41.01524107 + 1.5 Combine like terms: 41.01524107 + 1.5 = 42.51524107 t = 42.51524107 Simplifying t = 42.51524107Subproblem 2
t + -1.5 = -41.01524107 Simplifying t + -1.5 = -41.01524107 Reorder the terms: -1.5 + t = -41.01524107 Solving -1.5 + t = -41.01524107 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -41.01524107 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -41.01524107 + 1.5 t = -41.01524107 + 1.5 Combine like terms: -41.01524107 + 1.5 = -39.51524107 t = -39.51524107 Simplifying t = -39.51524107Solution
The solution to the problem is based on the solutions from the subproblems. t = {42.51524107, -39.51524107}Solution
t = {42.51524107, -39.51524107}
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