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Simplifying 546000 = 8t2 + 8t + 450 Reorder the terms: 546000 = 450 + 8t + 8t2 Solving 546000 = 450 + 8t + 8t2 Solving for variable 't'. Combine like terms: 546000 + -450 = 545550 545550 + -8t + -8t2 = 450 + 8t + 8t2 + -450 + -8t + -8t2 Reorder the terms: 545550 + -8t + -8t2 = 450 + -450 + 8t + -8t + 8t2 + -8t2 Combine like terms: 450 + -450 = 0 545550 + -8t + -8t2 = 0 + 8t + -8t + 8t2 + -8t2 545550 + -8t + -8t2 = 8t + -8t + 8t2 + -8t2 Combine like terms: 8t + -8t = 0 545550 + -8t + -8t2 = 0 + 8t2 + -8t2 545550 + -8t + -8t2 = 8t2 + -8t2 Combine like terms: 8t2 + -8t2 = 0 545550 + -8t + -8t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(272775 + -4t + -4t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(272775 + -4t + -4t2)' equal to zero and attempt to solve: Simplifying 272775 + -4t + -4t2 = 0 Solving 272775 + -4t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -68193.75 + t + t2 = 0 Move the constant term to the right: Add '68193.75' to each side of the equation. -68193.75 + t + 68193.75 + t2 = 0 + 68193.75 Reorder the terms: -68193.75 + 68193.75 + t + t2 = 0 + 68193.75 Combine like terms: -68193.75 + 68193.75 = 0.00 0.00 + t + t2 = 0 + 68193.75 t + t2 = 0 + 68193.75 Combine like terms: 0 + 68193.75 = 68193.75 t + t2 = 68193.75 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 68193.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 68193.75 + 0.25 Combine like terms: 68193.75 + 0.25 = 68194 1.25 + t2 = 68194 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 68194 Calculate the square root of the right side: 261.139809298 Break this problem into two subproblems by setting (t + 0.5) equal to 261.139809298 and -261.139809298.Subproblem 1
t + 0.5 = 261.139809298 Simplifying t + 0.5 = 261.139809298 Reorder the terms: 0.5 + t = 261.139809298 Solving 0.5 + t = 261.139809298 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 261.139809298 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 261.139809298 + -0.5 t = 261.139809298 + -0.5 Combine like terms: 261.139809298 + -0.5 = 260.639809298 t = 260.639809298 Simplifying t = 260.639809298Subproblem 2
t + 0.5 = -261.139809298 Simplifying t + 0.5 = -261.139809298 Reorder the terms: 0.5 + t = -261.139809298 Solving 0.5 + t = -261.139809298 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -261.139809298 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -261.139809298 + -0.5 t = -261.139809298 + -0.5 Combine like terms: -261.139809298 + -0.5 = -261.639809298 t = -261.639809298 Simplifying t = -261.639809298Solution
The solution to the problem is based on the solutions from the subproblems. t = {260.639809298, -261.639809298}Solution
t = {260.639809298, -261.639809298}
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