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Simplifying 0 = 16t2 + 14t + 2 Reorder the terms: 0 = 2 + 14t + 16t2 Solving 0 = 2 + 14t + 16t2 Solving for variable 't'. Combine like terms: 0 + -2 = -2 -2 + -14t + -16t2 = 2 + 14t + 16t2 + -2 + -14t + -16t2 Reorder the terms: -2 + -14t + -16t2 = 2 + -2 + 14t + -14t + 16t2 + -16t2 Combine like terms: 2 + -2 = 0 -2 + -14t + -16t2 = 0 + 14t + -14t + 16t2 + -16t2 -2 + -14t + -16t2 = 14t + -14t + 16t2 + -16t2 Combine like terms: 14t + -14t = 0 -2 + -14t + -16t2 = 0 + 16t2 + -16t2 -2 + -14t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -2 + -14t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(1 + 7t + 8t2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(1 + 7t + 8t2)' equal to zero and attempt to solve: Simplifying 1 + 7t + 8t2 = 0 Solving 1 + 7t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + 0.875t + t2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + 0.875t + -0.125 + t2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + 0.875t + t2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + 0.875t + t2 = 0 + -0.125 0.875t + t2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 0.875t + t2 = -0.125 The t term is 0.875t. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875t + 0.19140625 + t2 = -0.125 + 0.19140625 Reorder the terms: 0.19140625 + 0.875t + t2 = -0.125 + 0.19140625 Combine like terms: -0.125 + 0.19140625 = 0.06640625 0.19140625 + 0.875t + t2 = 0.06640625 Factor a perfect square on the left side: (t + 0.4375)(t + 0.4375) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (t + 0.4375) equal to 0.257694102 and -0.257694102.Subproblem 1
t + 0.4375 = 0.257694102 Simplifying t + 0.4375 = 0.257694102 Reorder the terms: 0.4375 + t = 0.257694102 Solving 0.4375 + t = 0.257694102 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + t = 0.257694102 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = 0.257694102 + -0.4375 t = 0.257694102 + -0.4375 Combine like terms: 0.257694102 + -0.4375 = -0.179805898 t = -0.179805898 Simplifying t = -0.179805898Subproblem 2
t + 0.4375 = -0.257694102 Simplifying t + 0.4375 = -0.257694102 Reorder the terms: 0.4375 + t = -0.257694102 Solving 0.4375 + t = -0.257694102 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + t = -0.257694102 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + t = -0.257694102 + -0.4375 t = -0.257694102 + -0.4375 Combine like terms: -0.257694102 + -0.4375 = -0.695194102 t = -0.695194102 Simplifying t = -0.695194102Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.179805898, -0.695194102}Solution
t = {-0.179805898, -0.695194102}
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