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Simplifying 10t = -16t2 + 12t + 8 Reorder the terms: 10t = 8 + 12t + -16t2 Solving 10t = 8 + 12t + -16t2 Solving for variable 't'. Reorder the terms: -8 + 10t + -12t + 16t2 = 8 + 12t + -16t2 + -8 + -12t + 16t2 Combine like terms: 10t + -12t = -2t -8 + -2t + 16t2 = 8 + 12t + -16t2 + -8 + -12t + 16t2 Reorder the terms: -8 + -2t + 16t2 = 8 + -8 + 12t + -12t + -16t2 + 16t2 Combine like terms: 8 + -8 = 0 -8 + -2t + 16t2 = 0 + 12t + -12t + -16t2 + 16t2 -8 + -2t + 16t2 = 12t + -12t + -16t2 + 16t2 Combine like terms: 12t + -12t = 0 -8 + -2t + 16t2 = 0 + -16t2 + 16t2 -8 + -2t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -8 + -2t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + -1t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + -1t + 8t2)' equal to zero and attempt to solve: Simplifying -4 + -1t + 8t2 = 0 Solving -4 + -1t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.5 + -0.125t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.125t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.125t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.125t + t2 = 0 + 0.5 -0.125t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.125t + t2 = 0.5 The t term is -0.125t. Take half its coefficient (-0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. -0.125t + 0.00390625 + t2 = 0.5 + 0.00390625 Reorder the terms: 0.00390625 + -0.125t + t2 = 0.5 + 0.00390625 Combine like terms: 0.5 + 0.00390625 = 0.50390625 0.00390625 + -0.125t + t2 = 0.50390625 Factor a perfect square on the left side: (t + -0.0625)(t + -0.0625) = 0.50390625 Calculate the square root of the right side: 0.709863543 Break this problem into two subproblems by setting (t + -0.0625) equal to 0.709863543 and -0.709863543.Subproblem 1
t + -0.0625 = 0.709863543 Simplifying t + -0.0625 = 0.709863543 Reorder the terms: -0.0625 + t = 0.709863543 Solving -0.0625 + t = 0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + t = 0.709863543 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + t = 0.709863543 + 0.0625 t = 0.709863543 + 0.0625 Combine like terms: 0.709863543 + 0.0625 = 0.772363543 t = 0.772363543 Simplifying t = 0.772363543Subproblem 2
t + -0.0625 = -0.709863543 Simplifying t + -0.0625 = -0.709863543 Reorder the terms: -0.0625 + t = -0.709863543 Solving -0.0625 + t = -0.709863543 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + t = -0.709863543 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + t = -0.709863543 + 0.0625 t = -0.709863543 + 0.0625 Combine like terms: -0.709863543 + 0.0625 = -0.647363543 t = -0.647363543 Simplifying t = -0.647363543Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.772363543, -0.647363543}Solution
t = {0.772363543, -0.647363543}
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