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Simplifying -16t2 + 114t + 6 = 0 Reorder the terms: 6 + 114t + -16t2 = 0 Solving 6 + 114t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 57t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 57t + -8t2)' equal to zero and attempt to solve: Simplifying 3 + 57t + -8t2 = 0 Solving 3 + 57t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.375 + -7.125t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -7.125t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -7.125t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -7.125t + t2 = 0 + 0.375 -7.125t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -7.125t + t2 = 0.375 The t term is -7.125t. Take half its coefficient (-3.5625). Square it (12.69140625) and add it to both sides. Add '12.69140625' to each side of the equation. -7.125t + 12.69140625 + t2 = 0.375 + 12.69140625 Reorder the terms: 12.69140625 + -7.125t + t2 = 0.375 + 12.69140625 Combine like terms: 0.375 + 12.69140625 = 13.06640625 12.69140625 + -7.125t + t2 = 13.06640625 Factor a perfect square on the left side: (t + -3.5625)(t + -3.5625) = 13.06640625 Calculate the square root of the right side: 3.614748435 Break this problem into two subproblems by setting (t + -3.5625) equal to 3.614748435 and -3.614748435.Subproblem 1
t + -3.5625 = 3.614748435 Simplifying t + -3.5625 = 3.614748435 Reorder the terms: -3.5625 + t = 3.614748435 Solving -3.5625 + t = 3.614748435 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5625' to each side of the equation. -3.5625 + 3.5625 + t = 3.614748435 + 3.5625 Combine like terms: -3.5625 + 3.5625 = 0.0000 0.0000 + t = 3.614748435 + 3.5625 t = 3.614748435 + 3.5625 Combine like terms: 3.614748435 + 3.5625 = 7.177248435 t = 7.177248435 Simplifying t = 7.177248435Subproblem 2
t + -3.5625 = -3.614748435 Simplifying t + -3.5625 = -3.614748435 Reorder the terms: -3.5625 + t = -3.614748435 Solving -3.5625 + t = -3.614748435 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5625' to each side of the equation. -3.5625 + 3.5625 + t = -3.614748435 + 3.5625 Combine like terms: -3.5625 + 3.5625 = 0.0000 0.0000 + t = -3.614748435 + 3.5625 t = -3.614748435 + 3.5625 Combine like terms: -3.614748435 + 3.5625 = -0.052248435 t = -0.052248435 Simplifying t = -0.052248435Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.177248435, -0.052248435}Solution
t = {7.177248435, -0.052248435}
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