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Simplifying 3 = 16t2 + 20t + 6 Reorder the terms: 3 = 6 + 20t + 16t2 Solving 3 = 6 + 20t + 16t2 Solving for variable 't'. Combine like terms: 3 + -6 = -3 -3 + -20t + -16t2 = 6 + 20t + 16t2 + -6 + -20t + -16t2 Reorder the terms: -3 + -20t + -16t2 = 6 + -6 + 20t + -20t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 -3 + -20t + -16t2 = 0 + 20t + -20t + 16t2 + -16t2 -3 + -20t + -16t2 = 20t + -20t + 16t2 + -16t2 Combine like terms: 20t + -20t = 0 -3 + -20t + -16t2 = 0 + 16t2 + -16t2 -3 + -20t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -3 + -20t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(3 + 20t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + 20t + 16t2)' equal to zero and attempt to solve: Simplifying 3 + 20t + 16t2 = 0 Solving 3 + 20t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + 1.25t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + 1.25t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + 1.25t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + 1.25t + t2 = 0 + -0.1875 1.25t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 1.25t + t2 = -0.1875 The t term is 1.25t. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25t + 0.390625 + t2 = -0.1875 + 0.390625 Reorder the terms: 0.390625 + 1.25t + t2 = -0.1875 + 0.390625 Combine like terms: -0.1875 + 0.390625 = 0.203125 0.390625 + 1.25t + t2 = 0.203125 Factor a perfect square on the left side: (t + 0.625)(t + 0.625) = 0.203125 Calculate the square root of the right side: 0.450693909 Break this problem into two subproblems by setting (t + 0.625) equal to 0.450693909 and -0.450693909.Subproblem 1
t + 0.625 = 0.450693909 Simplifying t + 0.625 = 0.450693909 Reorder the terms: 0.625 + t = 0.450693909 Solving 0.625 + t = 0.450693909 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = 0.450693909 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = 0.450693909 + -0.625 t = 0.450693909 + -0.625 Combine like terms: 0.450693909 + -0.625 = -0.174306091 t = -0.174306091 Simplifying t = -0.174306091Subproblem 2
t + 0.625 = -0.450693909 Simplifying t + 0.625 = -0.450693909 Reorder the terms: 0.625 + t = -0.450693909 Solving 0.625 + t = -0.450693909 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = -0.450693909 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = -0.450693909 + -0.625 t = -0.450693909 + -0.625 Combine like terms: -0.450693909 + -0.625 = -1.075693909 t = -1.075693909 Simplifying t = -1.075693909Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.174306091, -1.075693909}Solution
t = {-0.174306091, -1.075693909}
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