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Simplifying 15 = 1 + 16t + -4t2 Solving 15 = 1 + 16t + -4t2 Solving for variable 't'. Combine like terms: 15 + -1 = 14 14 + -16t + 4t2 = 1 + 16t + -4t2 + -1 + -16t + 4t2 Reorder the terms: 14 + -16t + 4t2 = 1 + -1 + 16t + -16t + -4t2 + 4t2 Combine like terms: 1 + -1 = 0 14 + -16t + 4t2 = 0 + 16t + -16t + -4t2 + 4t2 14 + -16t + 4t2 = 16t + -16t + -4t2 + 4t2 Combine like terms: 16t + -16t = 0 14 + -16t + 4t2 = 0 + -4t2 + 4t2 14 + -16t + 4t2 = -4t2 + 4t2 Combine like terms: -4t2 + 4t2 = 0 14 + -16t + 4t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -8t + 2t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -8t + 2t2)' equal to zero and attempt to solve: Simplifying 7 + -8t + 2t2 = 0 Solving 7 + -8t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + -4t + t2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + -4t + -3.5 + t2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + -4t + t2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + -4t + t2 = 0 + -3.5 -4t + t2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 -4t + t2 = -3.5 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 = -3.5 + 4 Reorder the terms: 4 + -4t + t2 = -3.5 + 4 Combine like terms: -3.5 + 4 = 0.5 4 + -4t + t2 = 0.5 Factor a perfect square on the left side: (t + -2)(t + -2) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (t + -2) equal to 0.707106781 and -0.707106781.Subproblem 1
t + -2 = 0.707106781 Simplifying t + -2 = 0.707106781 Reorder the terms: -2 + t = 0.707106781 Solving -2 + t = 0.707106781 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 = 0.707106781 + 2 Combine like terms: -2 + 2 = 0 0 + t = 0.707106781 + 2 t = 0.707106781 + 2 Combine like terms: 0.707106781 + 2 = 2.707106781 t = 2.707106781 Simplifying t = 2.707106781Subproblem 2
t + -2 = -0.707106781 Simplifying t + -2 = -0.707106781 Reorder the terms: -2 + t = -0.707106781 Solving -2 + t = -0.707106781 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 = -0.707106781 + 2 Combine like terms: -2 + 2 = 0 0 + t = -0.707106781 + 2 t = -0.707106781 + 2 Combine like terms: -0.707106781 + 2 = 1.292893219 t = 1.292893219 Simplifying t = 1.292893219Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.707106781, 1.292893219}Solution
t = {2.707106781, 1.292893219}
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