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Simplifying t = 16t2 + 28t + 4 Reorder the terms: t = 4 + 28t + 16t2 Solving t = 4 + 28t + 16t2 Solving for variable 't'. Reorder the terms: -4 + t + -28t + -16t2 = 4 + 28t + 16t2 + -4 + -28t + -16t2 Combine like terms: t + -28t = -27t -4 + -27t + -16t2 = 4 + 28t + 16t2 + -4 + -28t + -16t2 Reorder the terms: -4 + -27t + -16t2 = 4 + -4 + 28t + -28t + 16t2 + -16t2 Combine like terms: 4 + -4 = 0 -4 + -27t + -16t2 = 0 + 28t + -28t + 16t2 + -16t2 -4 + -27t + -16t2 = 28t + -28t + 16t2 + -16t2 Combine like terms: 28t + -28t = 0 -4 + -27t + -16t2 = 0 + 16t2 + -16t2 -4 + -27t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -4 + -27t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(4 + 27t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(4 + 27t + 16t2)' equal to zero and attempt to solve: Simplifying 4 + 27t + 16t2 = 0 Solving 4 + 27t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.25 + 1.6875t + t2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.6875t + -0.25 + t2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.6875t + t2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.6875t + t2 = 0 + -0.25 1.6875t + t2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.6875t + t2 = -0.25 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 = -0.25 + 0.7119140625 Reorder the terms: 0.7119140625 + 1.6875t + t2 = -0.25 + 0.7119140625 Combine like terms: -0.25 + 0.7119140625 = 0.4619140625 0.7119140625 + 1.6875t + t2 = 0.4619140625 Factor a perfect square on the left side: (t + 0.84375)(t + 0.84375) = 0.4619140625 Calculate the square root of the right side: 0.679642599 Break this problem into two subproblems by setting (t + 0.84375) equal to 0.679642599 and -0.679642599.Subproblem 1
t + 0.84375 = 0.679642599 Simplifying t + 0.84375 = 0.679642599 Reorder the terms: 0.84375 + t = 0.679642599 Solving 0.84375 + t = 0.679642599 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 = 0.679642599 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + t = 0.679642599 + -0.84375 t = 0.679642599 + -0.84375 Combine like terms: 0.679642599 + -0.84375 = -0.164107401 t = -0.164107401 Simplifying t = -0.164107401Subproblem 2
t + 0.84375 = -0.679642599 Simplifying t + 0.84375 = -0.679642599 Reorder the terms: 0.84375 + t = -0.679642599 Solving 0.84375 + t = -0.679642599 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 = -0.679642599 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + t = -0.679642599 + -0.84375 t = -0.679642599 + -0.84375 Combine like terms: -0.679642599 + -0.84375 = -1.523392599 t = -1.523392599 Simplifying t = -1.523392599Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.164107401, -1.523392599}Solution
t = {-0.164107401, -1.523392599}
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