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Simplifying -16t2 + 16t + 81 = 0 Reorder the terms: 81 + 16t + -16t2 = 0 Solving 81 + 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'. -5.0625 + -1t + t2 = 0 Move the constant term to the right: Add '5.0625' to each side of the equation. -5.0625 + -1t + 5.0625 + t2 = 0 + 5.0625 Reorder the terms: -5.0625 + 5.0625 + -1t + t2 = 0 + 5.0625 Combine like terms: -5.0625 + 5.0625 = 0.0000 0.0000 + -1t + t2 = 0 + 5.0625 -1t + t2 = 0 + 5.0625 Combine like terms: 0 + 5.0625 = 5.0625 -1t + t2 = 5.0625 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 = 5.0625 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 5.0625 + 0.25 Combine like terms: 5.0625 + 0.25 = 5.3125 0.25 + -1t + t2 = 5.3125 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 5.3125 Calculate the square root of the right side: 2.304886114 Break this problem into two subproblems by setting (t + -0.5) equal to 2.304886114 and -2.304886114.Subproblem 1
t + -0.5 = 2.304886114 Simplifying t + -0.5 = 2.304886114 Reorder the terms: -0.5 + t = 2.304886114 Solving -0.5 + t = 2.304886114 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 = 2.304886114 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 2.304886114 + 0.5 t = 2.304886114 + 0.5 Combine like terms: 2.304886114 + 0.5 = 2.804886114 t = 2.804886114 Simplifying t = 2.804886114Subproblem 2
t + -0.5 = -2.304886114 Simplifying t + -0.5 = -2.304886114 Reorder the terms: -0.5 + t = -2.304886114 Solving -0.5 + t = -2.304886114 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 = -2.304886114 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -2.304886114 + 0.5 t = -2.304886114 + 0.5 Combine like terms: -2.304886114 + 0.5 = -1.804886114 t = -1.804886114 Simplifying t = -1.804886114Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.804886114, -1.804886114}
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