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Simplifying 0 = (-16t2) + 8t + 12 Reorder the terms: 0 = 12 + 8t + (-16t2) Solving 0 = 12 + 8t + (-16t2) Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -8t + (16t2) = 12 + 8t + (-16t2) + -12 + -8t + (16t2) Reorder the terms: -12 + -8t + (16t2) = 12 + -12 + 8t + -8t + (-16t2) + (16t2) Combine like terms: 12 + -12 = 0 -12 + -8t + (16t2) = 0 + 8t + -8t + (-16t2) + (16t2) -12 + -8t + (16t2) = 8t + -8t + (-16t2) + (16t2) Combine like terms: 8t + -8t = 0 -12 + -8t + (16t2) = 0 + (-16t2) + (16t2) -12 + -8t + (16t2) = (-16t2) + (16t2) Combine like terms: (-16t2) + (16t2) = 0 -12 + -8t + (16t2) = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -2t + (4t2)) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -2t + (4t2))' equal to zero and attempt to solve: Simplifying -3 + -2t + (4t2) = 0 Solving -3 + -2t + (4t2) = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -0.5t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.5t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.5t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.5t + t2 = 0 + 0.75 -0.5t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.5t + t2 = 0.75 The t term is -0.5t. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5t + 0.0625 + t2 = 0.75 + 0.0625 Reorder the terms: 0.0625 + -0.5t + t2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + -0.5t + t2 = 0.8125 Factor a perfect square on the left side: ((t) + -0.25)((t) + -0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting ((t) + -0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
(t) + -0.25 = 0.901387819 Simplifying (t) + -0.25 = 0.901387819 t + -0.25 = 0.901387819 Reorder the terms: -0.25 + t = 0.901387819 Solving -0.25 + t = 0.901387819 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = 0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = 0.901387819 + 0.25 t = 0.901387819 + 0.25 Combine like terms: 0.901387819 + 0.25 = 1.151387819 t = 1.151387819 Simplifying t = 1.151387819Subproblem 2
(t) + -0.25 = -0.901387819 Simplifying (t) + -0.25 = -0.901387819 t + -0.25 = -0.901387819 Reorder the terms: -0.25 + t = -0.901387819 Solving -0.25 + t = -0.901387819 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = -0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = -0.901387819 + 0.25 t = -0.901387819 + 0.25 Combine like terms: -0.901387819 + 0.25 = -0.651387819 t = -0.651387819 Simplifying t = -0.651387819Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.151387819, -0.651387819}Solution
t = {1.151387819, -0.651387819}
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