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Simplifying 31770 = 33.755t2 + -1078.02t + 40376.569 Reorder the terms: 31770 = 40376.569 + -1078.02t + 33.755t2 Solving 31770 = 40376.569 + -1078.02t + 33.755t2 Solving for variable 't'. Combine like terms: 31770 + -40376.569 = -8606.569 -8606.569 + 1078.02t + -33.755t2 = 40376.569 + -1078.02t + 33.755t2 + -40376.569 + 1078.02t + -33.755t2 Reorder the terms: -8606.569 + 1078.02t + -33.755t2 = 40376.569 + -40376.569 + -1078.02t + 1078.02t + 33.755t2 + -33.755t2 Combine like terms: 40376.569 + -40376.569 = 0.000 -8606.569 + 1078.02t + -33.755t2 = 0.000 + -1078.02t + 1078.02t + 33.755t2 + -33.755t2 -8606.569 + 1078.02t + -33.755t2 = -1078.02t + 1078.02t + 33.755t2 + -33.755t2 Combine like terms: -1078.02t + 1078.02t = 0.00 -8606.569 + 1078.02t + -33.755t2 = 0.00 + 33.755t2 + -33.755t2 -8606.569 + 1078.02t + -33.755t2 = 33.755t2 + -33.755t2 Combine like terms: 33.755t2 + -33.755t2 = 0.000 -8606.569 + 1078.02t + -33.755t2 = 0.000 Begin completing the square. Divide all terms by -33.755 the coefficient of the squared term: Divide each side by '-33.755'. 254.9716783 + -31.93660198t + t2 = 0 Move the constant term to the right: Add '-254.9716783' to each side of the equation. 254.9716783 + -31.93660198t + -254.9716783 + t2 = 0 + -254.9716783 Reorder the terms: 254.9716783 + -254.9716783 + -31.93660198t + t2 = 0 + -254.9716783 Combine like terms: 254.9716783 + -254.9716783 = 0.0000000 0.0000000 + -31.93660198t + t2 = 0 + -254.9716783 -31.93660198t + t2 = 0 + -254.9716783 Combine like terms: 0 + -254.9716783 = -254.9716783 -31.93660198t + t2 = -254.9716783 The t term is -31.93660198t. Take half its coefficient (-15.96830099). Square it (254.9866365) and add it to both sides. Add '254.9866365' to each side of the equation. -31.93660198t + 254.9866365 + t2 = -254.9716783 + 254.9866365 Reorder the terms: 254.9866365 + -31.93660198t + t2 = -254.9716783 + 254.9866365 Combine like terms: -254.9716783 + 254.9866365 = 0.0149582 254.9866365 + -31.93660198t + t2 = 0.0149582 Factor a perfect square on the left side: (t + -15.96830099)(t + -15.96830099) = 0.0149582 Calculate the square root of the right side: 0.12230372 Break this problem into two subproblems by setting (t + -15.96830099) equal to 0.12230372 and -0.12230372.Subproblem 1
t + -15.96830099 = 0.12230372 Simplifying t + -15.96830099 = 0.12230372 Reorder the terms: -15.96830099 + t = 0.12230372 Solving -15.96830099 + t = 0.12230372 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15.96830099' to each side of the equation. -15.96830099 + 15.96830099 + t = 0.12230372 + 15.96830099 Combine like terms: -15.96830099 + 15.96830099 = 0.00000000 0.00000000 + t = 0.12230372 + 15.96830099 t = 0.12230372 + 15.96830099 Combine like terms: 0.12230372 + 15.96830099 = 16.09060471 t = 16.09060471 Simplifying t = 16.09060471Subproblem 2
t + -15.96830099 = -0.12230372 Simplifying t + -15.96830099 = -0.12230372 Reorder the terms: -15.96830099 + t = -0.12230372 Solving -15.96830099 + t = -0.12230372 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15.96830099' to each side of the equation. -15.96830099 + 15.96830099 + t = -0.12230372 + 15.96830099 Combine like terms: -15.96830099 + 15.96830099 = 0.00000000 0.00000000 + t = -0.12230372 + 15.96830099 t = -0.12230372 + 15.96830099 Combine like terms: -0.12230372 + 15.96830099 = 15.84599727 t = 15.84599727 Simplifying t = 15.84599727Solution
The solution to the problem is based on the solutions from the subproblems. t = {16.09060471, 15.84599727}
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