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Simplifying 0 = 16t2 + 27t + 10 Reorder the terms: 0 = 10 + 27t + 16t2 Solving 0 = 10 + 27t + 16t2 Solving for variable 't'. Combine like terms: 0 + -10 = -10 -10 + -27t + -16t2 = 10 + 27t + 16t2 + -10 + -27t + -16t2 Reorder the terms: -10 + -27t + -16t2 = 10 + -10 + 27t + -27t + 16t2 + -16t2 Combine like terms: 10 + -10 = 0 -10 + -27t + -16t2 = 0 + 27t + -27t + 16t2 + -16t2 -10 + -27t + -16t2 = 27t + -27t + 16t2 + -16t2 Combine like terms: 27t + -27t = 0 -10 + -27t + -16t2 = 0 + 16t2 + -16t2 -10 + -27t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -10 + -27t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(10 + 27t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(10 + 27t + 16t2)' equal to zero and attempt to solve: Simplifying 10 + 27t + 16t2 = 0 Solving 10 + 27t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.625 + 1.6875t + t2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + 1.6875t + -0.625 + t2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + 1.6875t + t2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + 1.6875t + t2 = 0 + -0.625 1.6875t + t2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 1.6875t + t2 = -0.625 The t term is 1.6875t. Take half its coefficient (0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. 1.6875t + 0.7119140625 + t2 = -0.625 + 0.7119140625 Reorder the terms: 0.7119140625 + 1.6875t + t2 = -0.625 + 0.7119140625 Combine like terms: -0.625 + 0.7119140625 = 0.0869140625 0.7119140625 + 1.6875t + t2 = 0.0869140625 Factor a perfect square on the left side: (t + 0.84375)(t + 0.84375) = 0.0869140625 Calculate the square root of the right side: 0.29481191 Break this problem into two subproblems by setting (t + 0.84375) equal to 0.29481191 and -0.29481191.Subproblem 1
t + 0.84375 = 0.29481191 Simplifying t + 0.84375 = 0.29481191 Reorder the terms: 0.84375 + t = 0.29481191 Solving 0.84375 + t = 0.29481191 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.84375' to each side of the equation. 0.84375 + -0.84375 + t = 0.29481191 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + t = 0.29481191 + -0.84375 t = 0.29481191 + -0.84375 Combine like terms: 0.29481191 + -0.84375 = -0.54893809 t = -0.54893809 Simplifying t = -0.54893809Subproblem 2
t + 0.84375 = -0.29481191 Simplifying t + 0.84375 = -0.29481191 Reorder the terms: 0.84375 + t = -0.29481191 Solving 0.84375 + t = -0.29481191 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.84375' to each side of the equation. 0.84375 + -0.84375 + t = -0.29481191 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + t = -0.29481191 + -0.84375 t = -0.29481191 + -0.84375 Combine like terms: -0.29481191 + -0.84375 = -1.13856191 t = -1.13856191 Simplifying t = -1.13856191Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.54893809, -1.13856191}Solution
t = {-0.54893809, -1.13856191}
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