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Simplifying -16t2 + -38t + 14 = 0 Reorder the terms: 14 + -38t + -16t2 = 0 Solving 14 + -38t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -19t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -19t + -8t2)' equal to zero and attempt to solve: Simplifying 7 + -19t + -8t2 = 0 Solving 7 + -19t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.875 + 2.375t + t2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 2.375t + 0.875 + t2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + 2.375t + t2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 2.375t + t2 = 0 + 0.875 2.375t + t2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 2.375t + t2 = 0.875 The t term is 2.375t. Take half its coefficient (1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. 2.375t + 1.41015625 + t2 = 0.875 + 1.41015625 Reorder the terms: 1.41015625 + 2.375t + t2 = 0.875 + 1.41015625 Combine like terms: 0.875 + 1.41015625 = 2.28515625 1.41015625 + 2.375t + t2 = 2.28515625 Factor a perfect square on the left side: (t + 1.1875)(t + 1.1875) = 2.28515625 Calculate the square root of the right side: 1.511673328 Break this problem into two subproblems by setting (t + 1.1875) equal to 1.511673328 and -1.511673328.Subproblem 1
t + 1.1875 = 1.511673328 Simplifying t + 1.1875 = 1.511673328 Reorder the terms: 1.1875 + t = 1.511673328 Solving 1.1875 + t = 1.511673328 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + t = 1.511673328 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + t = 1.511673328 + -1.1875 t = 1.511673328 + -1.1875 Combine like terms: 1.511673328 + -1.1875 = 0.324173328 t = 0.324173328 Simplifying t = 0.324173328Subproblem 2
t + 1.1875 = -1.511673328 Simplifying t + 1.1875 = -1.511673328 Reorder the terms: 1.1875 + t = -1.511673328 Solving 1.1875 + t = -1.511673328 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + t = -1.511673328 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + t = -1.511673328 + -1.1875 t = -1.511673328 + -1.1875 Combine like terms: -1.511673328 + -1.1875 = -2.699173328 t = -2.699173328 Simplifying t = -2.699173328Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.324173328, -2.699173328}Solution
t = {0.324173328, -2.699173328}
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