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Simplifying -162t2 + 112t + 128 = 112 Reorder the terms: 128 + 112t + -162t2 = 112 Solving 128 + 112t + -162t2 = 112 Solving for variable 't'. Reorder the terms: 128 + -112 + 112t + -162t2 = 112 + -112 Combine like terms: 128 + -112 = 16 16 + 112t + -162t2 = 112 + -112 Combine like terms: 112 + -112 = 0 16 + 112t + -162t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + 56t + -81t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + 56t + -81t2)' equal to zero and attempt to solve: Simplifying 8 + 56t + -81t2 = 0 Solving 8 + 56t + -81t2 = 0 Begin completing the square. Divide all terms by -81 the coefficient of the squared term: Divide each side by '-81'. -0.0987654321 + -0.6913580247t + t2 = 0 Move the constant term to the right: Add '0.0987654321' to each side of the equation. -0.0987654321 + -0.6913580247t + 0.0987654321 + t2 = 0 + 0.0987654321 Reorder the terms: -0.0987654321 + 0.0987654321 + -0.6913580247t + t2 = 0 + 0.0987654321 Combine like terms: -0.0987654321 + 0.0987654321 = 0.0000000000 0.0000000000 + -0.6913580247t + t2 = 0 + 0.0987654321 -0.6913580247t + t2 = 0 + 0.0987654321 Combine like terms: 0 + 0.0987654321 = 0.0987654321 -0.6913580247t + t2 = 0.0987654321 The t term is -0.6913580247t. Take half its coefficient (-0.3456790124). Square it (0.1194939796) and add it to both sides. Add '0.1194939796' to each side of the equation. -0.6913580247t + 0.1194939796 + t2 = 0.0987654321 + 0.1194939796 Reorder the terms: 0.1194939796 + -0.6913580247t + t2 = 0.0987654321 + 0.1194939796 Combine like terms: 0.0987654321 + 0.1194939796 = 0.2182594117 0.1194939796 + -0.6913580247t + t2 = 0.2182594117 Factor a perfect square on the left side: (t + -0.3456790124)(t + -0.3456790124) = 0.2182594117 Calculate the square root of the right side: 0.467182418 Break this problem into two subproblems by setting (t + -0.3456790124) equal to 0.467182418 and -0.467182418.Subproblem 1
t + -0.3456790124 = 0.467182418 Simplifying t + -0.3456790124 = 0.467182418 Reorder the terms: -0.3456790124 + t = 0.467182418 Solving -0.3456790124 + t = 0.467182418 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3456790124' to each side of the equation. -0.3456790124 + 0.3456790124 + t = 0.467182418 + 0.3456790124 Combine like terms: -0.3456790124 + 0.3456790124 = 0.0000000000 0.0000000000 + t = 0.467182418 + 0.3456790124 t = 0.467182418 + 0.3456790124 Combine like terms: 0.467182418 + 0.3456790124 = 0.8128614304 t = 0.8128614304 Simplifying t = 0.8128614304Subproblem 2
t + -0.3456790124 = -0.467182418 Simplifying t + -0.3456790124 = -0.467182418 Reorder the terms: -0.3456790124 + t = -0.467182418 Solving -0.3456790124 + t = -0.467182418 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3456790124' to each side of the equation. -0.3456790124 + 0.3456790124 + t = -0.467182418 + 0.3456790124 Combine like terms: -0.3456790124 + 0.3456790124 = 0.0000000000 0.0000000000 + t = -0.467182418 + 0.3456790124 t = -0.467182418 + 0.3456790124 Combine like terms: -0.467182418 + 0.3456790124 = -0.1215034056 t = -0.1215034056 Simplifying t = -0.1215034056Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.8128614304, -0.1215034056}Solution
t = {0.8128614304, -0.1215034056}
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