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Simplifying -16t2 + 16t + 65 = 0 Reorder the terms: 65 + 16t + -16t2 = 0 Solving 65 + 16t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -4.0625 + -1t + t2 = 0 Move the constant term to the right: Add '4.0625' to each side of the equation. -4.0625 + -1t + 4.0625 + t2 = 0 + 4.0625 Reorder the terms: -4.0625 + 4.0625 + -1t + t2 = 0 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + -1t + t2 = 0 + 4.0625 -1t + t2 = 0 + 4.0625 Combine like terms: 0 + 4.0625 = 4.0625 -1t + t2 = 4.0625 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 4.0625 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 4.0625 + 0.25 Combine like terms: 4.0625 + 0.25 = 4.3125 0.25 + -1t + t2 = 4.3125 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 4.3125 Calculate the square root of the right side: 2.076655966 Break this problem into two subproblems by setting (t + -0.5) equal to 2.076655966 and -2.076655966.Subproblem 1
t + -0.5 = 2.076655966 Simplifying t + -0.5 = 2.076655966 Reorder the terms: -0.5 + t = 2.076655966 Solving -0.5 + t = 2.076655966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 2.076655966 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 2.076655966 + 0.5 t = 2.076655966 + 0.5 Combine like terms: 2.076655966 + 0.5 = 2.576655966 t = 2.576655966 Simplifying t = 2.576655966Subproblem 2
t + -0.5 = -2.076655966 Simplifying t + -0.5 = -2.076655966 Reorder the terms: -0.5 + t = -2.076655966 Solving -0.5 + t = -2.076655966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -2.076655966 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -2.076655966 + 0.5 t = -2.076655966 + 0.5 Combine like terms: -2.076655966 + 0.5 = -1.576655966 t = -1.576655966 Simplifying t = -1.576655966Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.576655966, -1.576655966}
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