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Simplifying 0 = 16t2 + 64t + 8 Reorder the terms: 0 = 8 + 64t + 16t2 Solving 0 = 8 + 64t + 16t2 Solving for variable 't'. Combine like terms: 0 + -8 = -8 -8 + -64t + -16t2 = 8 + 64t + 16t2 + -8 + -64t + -16t2 Reorder the terms: -8 + -64t + -16t2 = 8 + -8 + 64t + -64t + 16t2 + -16t2 Combine like terms: 8 + -8 = 0 -8 + -64t + -16t2 = 0 + 64t + -64t + 16t2 + -16t2 -8 + -64t + -16t2 = 64t + -64t + 16t2 + -16t2 Combine like terms: 64t + -64t = 0 -8 + -64t + -16t2 = 0 + 16t2 + -16t2 -8 + -64t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -8 + -64t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-8'. -8(1 + 8t + 2t2) = 0 Ignore the factor -8.Subproblem 1
Set the factor '(1 + 8t + 2t2)' equal to zero and attempt to solve: Simplifying 1 + 8t + 2t2 = 0 Solving 1 + 8t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 4t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 4t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 4t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 4t + t2 = 0 + -0.5 4t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 4t + t2 = -0.5 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 = -0.5 + 4 Reorder the terms: 4 + 4t + t2 = -0.5 + 4 Combine like terms: -0.5 + 4 = 3.5 4 + 4t + t2 = 3.5 Factor a perfect square on the left side: (t + 2)(t + 2) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (t + 2) equal to 1.870828693 and -1.870828693.Subproblem 1
t + 2 = 1.870828693 Simplifying t + 2 = 1.870828693 Reorder the terms: 2 + t = 1.870828693 Solving 2 + t = 1.870828693 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 = 1.870828693 + -2 Combine like terms: 2 + -2 = 0 0 + t = 1.870828693 + -2 t = 1.870828693 + -2 Combine like terms: 1.870828693 + -2 = -0.129171307 t = -0.129171307 Simplifying t = -0.129171307Subproblem 2
t + 2 = -1.870828693 Simplifying t + 2 = -1.870828693 Reorder the terms: 2 + t = -1.870828693 Solving 2 + t = -1.870828693 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 = -1.870828693 + -2 Combine like terms: 2 + -2 = 0 0 + t = -1.870828693 + -2 t = -1.870828693 + -2 Combine like terms: -1.870828693 + -2 = -3.870828693 t = -3.870828693 Simplifying t = -3.870828693Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.129171307, -3.870828693}Solution
t = {-0.129171307, -3.870828693}
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