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Simplifying -1T2 + -1T = 6T3 Reorder the terms: -1T + -1T2 = 6T3 Solving -1T + -1T2 = 6T3 Solving for variable 'T'. Combine like terms: 6T3 + -6T3 = 0 -1T + -1T2 + -6T3 = 0 Factor out the Greatest Common Factor (GCF), '-1T'. -1T(1 + T + 6T2) = 0 Ignore the factor -1.Subproblem 1
Set the factor 'T' equal to zero and attempt to solve: Simplifying T = 0 Solving T = 0 Move all terms containing T to the left, all other terms to the right. Simplifying T = 0Subproblem 2
Set the factor '(1 + T + 6T2)' equal to zero and attempt to solve: Simplifying 1 + T + 6T2 = 0 Solving 1 + T + 6T2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 0.1666666667T + T2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 0.1666666667T + -0.1666666667 + T2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 0.1666666667T + T2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 0.1666666667T + T2 = 0 + -0.1666666667 0.1666666667T + T2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 0.1666666667T + T2 = -0.1666666667 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. 0.1666666667T + 0.25 + T2 = -0.1666666667 + 0.25 Reorder the terms: 0.25 + 0.1666666667T + T2 = -0.1666666667 + 0.25 Combine like terms: -0.1666666667 + 0.25 = 0.0833333333 0.25 + 0.1666666667T + T2 = 0.0833333333 Factor a perfect square on the left side: (T + 0.5)(T + 0.5) = 0.0833333333 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (T + 0.5) equal to 0.288675135 and -0.288675135.Subproblem 1
T + 0.5 = 0.288675135 Simplifying T + 0.5 = 0.288675135 Reorder the terms: 0.5 + T = 0.288675135 Solving 0.5 + T = 0.288675135 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 = 0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + T = 0.288675135 + -0.5 T = 0.288675135 + -0.5 Combine like terms: 0.288675135 + -0.5 = -0.211324865 T = -0.211324865 Simplifying T = -0.211324865Subproblem 2
T + 0.5 = -0.288675135 Simplifying T + 0.5 = -0.288675135 Reorder the terms: 0.5 + T = -0.288675135 Solving 0.5 + T = -0.288675135 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 = -0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + T = -0.288675135 + -0.5 T = -0.288675135 + -0.5 Combine like terms: -0.288675135 + -0.5 = -0.788675135 T = -0.788675135 Simplifying T = -0.788675135Solution
The solution to the problem is based on the solutions from the subproblems. T = {-0.211324865, -0.788675135}Solution
T = {0, -0.211324865, -0.788675135}
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