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Simplifying 16t2 + 66t + 40 = 0 Reorder the terms: 40 + 66t + 16t2 = 0 Solving 40 + 66t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(20 + 33t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(20 + 33t + 8t2)' equal to zero and attempt to solve: Simplifying 20 + 33t + 8t2 = 0 Solving 20 + 33t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 2.5 + 4.125t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 4.125t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 4.125t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 4.125t + t2 = 0 + -2.5 4.125t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 4.125t + t2 = -2.5 The t term is 4.125t. Take half its coefficient (2.0625). Square it (4.25390625) and add it to both sides. Add '4.25390625' to each side of the equation. 4.125t + 4.25390625 + t2 = -2.5 + 4.25390625 Reorder the terms: 4.25390625 + 4.125t + t2 = -2.5 + 4.25390625 Combine like terms: -2.5 + 4.25390625 = 1.75390625 4.25390625 + 4.125t + t2 = 1.75390625 Factor a perfect square on the left side: (t + 2.0625)(t + 2.0625) = 1.75390625 Calculate the square root of the right side: 1.324351256 Break this problem into two subproblems by setting (t + 2.0625) equal to 1.324351256 and -1.324351256.Subproblem 1
t + 2.0625 = 1.324351256 Simplifying t + 2.0625 = 1.324351256 Reorder the terms: 2.0625 + t = 1.324351256 Solving 2.0625 + t = 1.324351256 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.0625' to each side of the equation. 2.0625 + -2.0625 + t = 1.324351256 + -2.0625 Combine like terms: 2.0625 + -2.0625 = 0.0000 0.0000 + t = 1.324351256 + -2.0625 t = 1.324351256 + -2.0625 Combine like terms: 1.324351256 + -2.0625 = -0.738148744 t = -0.738148744 Simplifying t = -0.738148744Subproblem 2
t + 2.0625 = -1.324351256 Simplifying t + 2.0625 = -1.324351256 Reorder the terms: 2.0625 + t = -1.324351256 Solving 2.0625 + t = -1.324351256 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.0625' to each side of the equation. 2.0625 + -2.0625 + t = -1.324351256 + -2.0625 Combine like terms: 2.0625 + -2.0625 = 0.0000 0.0000 + t = -1.324351256 + -2.0625 t = -1.324351256 + -2.0625 Combine like terms: -1.324351256 + -2.0625 = -3.386851256 t = -3.386851256 Simplifying t = -3.386851256Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.738148744, -3.386851256}Solution
t = {-0.738148744, -3.386851256}
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