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Simplifying 180 = 14t2 + -100t Reorder the terms: 180 = -100t + 14t2 Solving 180 = -100t + 14t2 Solving for variable 't'. Reorder the terms: 180 + 100t + -14t2 = -100t + 100t + 14t2 + -14t2 Combine like terms: -100t + 100t = 0 180 + 100t + -14t2 = 0 + 14t2 + -14t2 180 + 100t + -14t2 = 14t2 + -14t2 Combine like terms: 14t2 + -14t2 = 0 180 + 100t + -14t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(90 + 50t + -7t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + 50t + -7t2)' equal to zero and attempt to solve: Simplifying 90 + 50t + -7t2 = 0 Solving 90 + 50t + -7t2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -12.85714286 + -7.142857143t + t2 = 0 Move the constant term to the right: Add '12.85714286' to each side of the equation. -12.85714286 + -7.142857143t + 12.85714286 + t2 = 0 + 12.85714286 Reorder the terms: -12.85714286 + 12.85714286 + -7.142857143t + t2 = 0 + 12.85714286 Combine like terms: -12.85714286 + 12.85714286 = 0.00000000 0.00000000 + -7.142857143t + t2 = 0 + 12.85714286 -7.142857143t + t2 = 0 + 12.85714286 Combine like terms: 0 + 12.85714286 = 12.85714286 -7.142857143t + t2 = 12.85714286 The t term is -7.142857143t. Take half its coefficient (-3.571428572). Square it (12.75510204) and add it to both sides. Add '12.75510204' to each side of the equation. -7.142857143t + 12.75510204 + t2 = 12.85714286 + 12.75510204 Reorder the terms: 12.75510204 + -7.142857143t + t2 = 12.85714286 + 12.75510204 Combine like terms: 12.85714286 + 12.75510204 = 25.6122449 12.75510204 + -7.142857143t + t2 = 25.6122449 Factor a perfect square on the left side: (t + -3.571428572)(t + -3.571428572) = 25.6122449 Calculate the square root of the right side: 5.060854167 Break this problem into two subproblems by setting (t + -3.571428572) equal to 5.060854167 and -5.060854167.Subproblem 1
t + -3.571428572 = 5.060854167 Simplifying t + -3.571428572 = 5.060854167 Reorder the terms: -3.571428572 + t = 5.060854167 Solving -3.571428572 + t = 5.060854167 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.571428572' to each side of the equation. -3.571428572 + 3.571428572 + t = 5.060854167 + 3.571428572 Combine like terms: -3.571428572 + 3.571428572 = 0.000000000 0.000000000 + t = 5.060854167 + 3.571428572 t = 5.060854167 + 3.571428572 Combine like terms: 5.060854167 + 3.571428572 = 8.632282739 t = 8.632282739 Simplifying t = 8.632282739Subproblem 2
t + -3.571428572 = -5.060854167 Simplifying t + -3.571428572 = -5.060854167 Reorder the terms: -3.571428572 + t = -5.060854167 Solving -3.571428572 + t = -5.060854167 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.571428572' to each side of the equation. -3.571428572 + 3.571428572 + t = -5.060854167 + 3.571428572 Combine like terms: -3.571428572 + 3.571428572 = 0.000000000 0.000000000 + t = -5.060854167 + 3.571428572 t = -5.060854167 + 3.571428572 Combine like terms: -5.060854167 + 3.571428572 = -1.489425595 t = -1.489425595 Simplifying t = -1.489425595Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.632282739, -1.489425595}Solution
t = {8.632282739, -1.489425595}
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