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Simplifying (19t3 + -5t2) + -1(7t3 + -3t) + (15t3 + -8t) = 0 Reorder the terms: (-5t2 + 19t3) + -1(7t3 + -3t) + (15t3 + -8t) = 0 Remove parenthesis around (-5t2 + 19t3) -5t2 + 19t3 + -1(7t3 + -3t) + (15t3 + -8t) = 0 Reorder the terms: -5t2 + 19t3 + -1(-3t + 7t3) + (15t3 + -8t) = 0 -5t2 + 19t3 + (-3t * -1 + 7t3 * -1) + (15t3 + -8t) = 0 -5t2 + 19t3 + (3t + -7t3) + (15t3 + -8t) = 0 Reorder the terms: -5t2 + 19t3 + 3t + -7t3 + (-8t + 15t3) = 0 Remove parenthesis around (-8t + 15t3) -5t2 + 19t3 + 3t + -7t3 + -8t + 15t3 = 0 Reorder the terms: 3t + -8t + -5t2 + 19t3 + -7t3 + 15t3 = 0 Combine like terms: 3t + -8t = -5t -5t + -5t2 + 19t3 + -7t3 + 15t3 = 0 Combine like terms: 19t3 + -7t3 = 12t3 -5t + -5t2 + 12t3 + 15t3 = 0 Combine like terms: 12t3 + 15t3 = 27t3 -5t + -5t2 + 27t3 = 0 Solving -5t + -5t2 + 27t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't'. t(-5 + -5t + 27t2) = 0Subproblem 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 '(-5 + -5t + 27t2)' equal to zero and attempt to solve: Simplifying -5 + -5t + 27t2 = 0 Solving -5 + -5t + 27t2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.1851851852 + -0.1851851852t + t2 = 0 Move the constant term to the right: Add '0.1851851852' to each side of the equation. -0.1851851852 + -0.1851851852t + 0.1851851852 + t2 = 0 + 0.1851851852 Reorder the terms: -0.1851851852 + 0.1851851852 + -0.1851851852t + t2 = 0 + 0.1851851852 Combine like terms: -0.1851851852 + 0.1851851852 = 0.0000000000 0.0000000000 + -0.1851851852t + t2 = 0 + 0.1851851852 -0.1851851852t + t2 = 0 + 0.1851851852 Combine like terms: 0 + 0.1851851852 = 0.1851851852 -0.1851851852t + t2 = 0.1851851852 The t term is -0.1851851852t. Take half its coefficient (-0.0925925926). Square it (0.008573388204) and add it to both sides. Add '0.008573388204' to each side of the equation. -0.1851851852t + 0.008573388204 + t2 = 0.1851851852 + 0.008573388204 Reorder the terms: 0.008573388204 + -0.1851851852t + t2 = 0.1851851852 + 0.008573388204 Combine like terms: 0.1851851852 + 0.008573388204 = 0.193758573404 0.008573388204 + -0.1851851852t + t2 = 0.193758573404 Factor a perfect square on the left side: (t + -0.0925925926)(t + -0.0925925926) = 0.193758573404 Calculate the square root of the right side: 0.44018016 Break this problem into two subproblems by setting (t + -0.0925925926) equal to 0.44018016 and -0.44018016.Subproblem 1
t + -0.0925925926 = 0.44018016 Simplifying t + -0.0925925926 = 0.44018016 Reorder the terms: -0.0925925926 + t = 0.44018016 Solving -0.0925925926 + t = 0.44018016 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.0925925926' to each side of the equation. -0.0925925926 + 0.0925925926 + t = 0.44018016 + 0.0925925926 Combine like terms: -0.0925925926 + 0.0925925926 = 0.0000000000 0.0000000000 + t = 0.44018016 + 0.0925925926 t = 0.44018016 + 0.0925925926 Combine like terms: 0.44018016 + 0.0925925926 = 0.5327727526 t = 0.5327727526 Simplifying t = 0.5327727526Subproblem 2
t + -0.0925925926 = -0.44018016 Simplifying t + -0.0925925926 = -0.44018016 Reorder the terms: -0.0925925926 + t = -0.44018016 Solving -0.0925925926 + t = -0.44018016 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.0925925926' to each side of the equation. -0.0925925926 + 0.0925925926 + t = -0.44018016 + 0.0925925926 Combine like terms: -0.0925925926 + 0.0925925926 = 0.0000000000 0.0000000000 + t = -0.44018016 + 0.0925925926 t = -0.44018016 + 0.0925925926 Combine like terms: -0.44018016 + 0.0925925926 = -0.3475875674 t = -0.3475875674 Simplifying t = -0.3475875674Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.5327727526, -0.3475875674}Solution
t = {0, 0.5327727526, -0.3475875674}
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