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Simplifying 12t2 + 2t + -16 = 0 Reorder the terms: -16 + 2t + 12t2 = 0 Solving -16 + 2t + 12t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-8 + t + 6t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-8 + t + 6t2)' equal to zero and attempt to solve: Simplifying -8 + t + 6t2 = 0 Solving -8 + t + 6t2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 0.1666666667t + t2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 0.1666666667t + 1.333333333 + t2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 0.1666666667t + t2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 0.1666666667t + t2 = 0 + 1.333333333 0.1666666667t + t2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 0.1666666667t + t2 = 1.333333333 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1666666667t + 0.25 + t2 = 1.333333333 + 0.25 Reorder the terms: 0.25 + 0.1666666667t + t2 = 1.333333333 + 0.25 Combine like terms: 1.333333333 + 0.25 = 1.583333333 0.25 + 0.1666666667t + t2 = 1.583333333 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (t + 0.5) equal to 1.258305739 and -1.258305739.Subproblem 1
t + 0.5 = 1.258305739 Simplifying t + 0.5 = 1.258305739 Reorder the terms: 0.5 + t = 1.258305739 Solving 0.5 + t = 1.258305739 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.258305739 + -0.5 t = 1.258305739 + -0.5 Combine like terms: 1.258305739 + -0.5 = 0.758305739 t = 0.758305739 Simplifying t = 0.758305739Subproblem 2
t + 0.5 = -1.258305739 Simplifying t + 0.5 = -1.258305739 Reorder the terms: 0.5 + t = -1.258305739 Solving 0.5 + t = -1.258305739 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.258305739 + -0.5 t = -1.258305739 + -0.5 Combine like terms: -1.258305739 + -0.5 = -1.758305739 t = -1.758305739 Simplifying t = -1.758305739Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.758305739, -1.758305739}Solution
t = {0.758305739, -1.758305739}
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