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Simplifying -9t2 + -5t + 5t3 + -1t3 + 12t2 + -3t = 0 Reorder the terms: -5t + -3t + -9t2 + 12t2 + 5t3 + -1t3 = 0 Combine like terms: -5t + -3t = -8t -8t + -9t2 + 12t2 + 5t3 + -1t3 = 0 Combine like terms: -9t2 + 12t2 = 3t2 -8t + 3t2 + 5t3 + -1t3 = 0 Combine like terms: 5t3 + -1t3 = 4t3 -8t + 3t2 + 4t3 = 0 Solving -8t + 3t2 + 4t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't'. t(-8 + 3t + 4t2) = 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 '(-8 + 3t + 4t2)' equal to zero and attempt to solve: Simplifying -8 + 3t + 4t2 = 0 Solving -8 + 3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2 + 0.75t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.75t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 0.75t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.75t + t2 = 0 + 2 0.75t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.75t + t2 = 2 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 2 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 2 + 0.140625 Combine like terms: 2 + 0.140625 = 2.140625 0.140625 + 0.75t + t2 = 2.140625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 2.140625 Calculate the square root of the right side: 1.463087489 Break this problem into two subproblems by setting (t + 0.375) equal to 1.463087489 and -1.463087489.Subproblem 1
t + 0.375 = 1.463087489 Simplifying t + 0.375 = 1.463087489 Reorder the terms: 0.375 + t = 1.463087489 Solving 0.375 + t = 1.463087489 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.463087489 + -0.375 t = 1.463087489 + -0.375 Combine like terms: 1.463087489 + -0.375 = 1.088087489 t = 1.088087489 Simplifying t = 1.088087489Subproblem 2
t + 0.375 = -1.463087489 Simplifying t + 0.375 = -1.463087489 Reorder the terms: 0.375 + t = -1.463087489 Solving 0.375 + t = -1.463087489 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.463087489 + -0.375 t = -1.463087489 + -0.375 Combine like terms: -1.463087489 + -0.375 = -1.838087489 t = -1.838087489 Simplifying t = -1.838087489Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.088087489, -1.838087489}Solution
t = {0, 1.088087489, -1.838087489}
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