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Simplifying 0 = -64 + -16t2 + 64t + 12 Reorder the terms: 0 = -64 + 12 + 64t + -16t2 Combine like terms: -64 + 12 = -52 0 = -52 + 64t + -16t2 Solving 0 = -52 + 64t + -16t2 Solving for variable 't'. Combine like terms: 0 + 52 = 52 52 + -64t + 16t2 = -52 + 64t + -16t2 + 52 + -64t + 16t2 Reorder the terms: 52 + -64t + 16t2 = -52 + 52 + 64t + -64t + -16t2 + 16t2 Combine like terms: -52 + 52 = 0 52 + -64t + 16t2 = 0 + 64t + -64t + -16t2 + 16t2 52 + -64t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 52 + -64t + 16t2 = 0 + -16t2 + 16t2 52 + -64t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 52 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(13 + -16t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(13 + -16t + 4t2)' equal to zero and attempt to solve: Simplifying 13 + -16t + 4t2 = 0 Solving 13 + -16t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 3.25 + -4t + t2 = 0 Move the constant term to the right: Add '-3.25' to each side of the equation. 3.25 + -4t + -3.25 + t2 = 0 + -3.25 Reorder the terms: 3.25 + -3.25 + -4t + t2 = 0 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + -4t + t2 = 0 + -3.25 -4t + t2 = 0 + -3.25 Combine like terms: 0 + -3.25 = -3.25 -4t + t2 = -3.25 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -3.25 + 4 Reorder the terms: 4 + -4t + t2 = -3.25 + 4 Combine like terms: -3.25 + 4 = 0.75 4 + -4t + t2 = 0.75 Factor a perfect square on the left side: (t + -2)(t + -2) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (t + -2) equal to 0.866025404 and -0.866025404.Subproblem 1
t + -2 = 0.866025404 Simplifying t + -2 = 0.866025404 Reorder the terms: -2 + t = 0.866025404 Solving -2 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + t = 0.866025404 + 2 t = 0.866025404 + 2 Combine like terms: 0.866025404 + 2 = 2.866025404 t = 2.866025404 Simplifying t = 2.866025404Subproblem 2
t + -2 = -0.866025404 Simplifying t + -2 = -0.866025404 Reorder the terms: -2 + t = -0.866025404 Solving -2 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + t = -0.866025404 + 2 t = -0.866025404 + 2 Combine like terms: -0.866025404 + 2 = 1.133974596 t = 1.133974596 Simplifying t = 1.133974596Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.866025404, 1.133974596}Solution
t = {2.866025404, 1.133974596}
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