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Simplifying 84t2 + -12t + -24 = 0 Reorder the terms: -24 + -12t + 84t2 = 0 Solving -24 + -12t + 84t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '12'. 12(-2 + -1t + 7t2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(-2 + -1t + 7t2)' equal to zero and attempt to solve: Simplifying -2 + -1t + 7t2 = 0 Solving -2 + -1t + 7t2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.2857142857 + -0.1428571429t + t2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -0.1428571429t + 0.2857142857 + t2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -0.1428571429t + t2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -0.1428571429t + t2 = 0 + 0.2857142857 -0.1428571429t + t2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -0.1428571429t + t2 = 0.2857142857 The t term is -0.1428571429t. Take half its coefficient (-0.07142857145). Square it (0.005102040819) and add it to both sides. Add '0.005102040819' to each side of the equation. -0.1428571429t + 0.005102040819 + t2 = 0.2857142857 + 0.005102040819 Reorder the terms: 0.005102040819 + -0.1428571429t + t2 = 0.2857142857 + 0.005102040819 Combine like terms: 0.2857142857 + 0.005102040819 = 0.290816326519 0.005102040819 + -0.1428571429t + t2 = 0.290816326519 Factor a perfect square on the left side: (t + -0.07142857145)(t + -0.07142857145) = 0.290816326519 Calculate the square root of the right side: 0.539273888 Break this problem into two subproblems by setting (t + -0.07142857145) equal to 0.539273888 and -0.539273888.Subproblem 1
t + -0.07142857145 = 0.539273888 Simplifying t + -0.07142857145 = 0.539273888 Reorder the terms: -0.07142857145 + t = 0.539273888 Solving -0.07142857145 + t = 0.539273888 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + t = 0.539273888 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + t = 0.539273888 + 0.07142857145 t = 0.539273888 + 0.07142857145 Combine like terms: 0.539273888 + 0.07142857145 = 0.61070245945 t = 0.61070245945 Simplifying t = 0.61070245945Subproblem 2
t + -0.07142857145 = -0.539273888 Simplifying t + -0.07142857145 = -0.539273888 Reorder the terms: -0.07142857145 + t = -0.539273888 Solving -0.07142857145 + t = -0.539273888 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + t = -0.539273888 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + t = -0.539273888 + 0.07142857145 t = -0.539273888 + 0.07142857145 Combine like terms: -0.539273888 + 0.07142857145 = -0.46784531655 t = -0.46784531655 Simplifying t = -0.46784531655Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.61070245945, -0.46784531655}Solution
t = {0.61070245945, -0.46784531655}
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