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Simplifying 8.125 = -5t2 + 15t + 5 Reorder the terms: 8.125 = 5 + 15t + -5t2 Solving 8.125 = 5 + 15t + -5t2 Solving for variable 't'. Combine like terms: 8.125 + -5 = 3.125 3.125 + -15t + 5t2 = 5 + 15t + -5t2 + -5 + -15t + 5t2 Reorder the terms: 3.125 + -15t + 5t2 = 5 + -5 + 15t + -15t + -5t2 + 5t2 Combine like terms: 5 + -5 = 0 3.125 + -15t + 5t2 = 0 + 15t + -15t + -5t2 + 5t2 3.125 + -15t + 5t2 = 15t + -15t + -5t2 + 5t2 Combine like terms: 15t + -15t = 0 3.125 + -15t + 5t2 = 0 + -5t2 + 5t2 3.125 + -15t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 3.125 + -15t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.625 + -3t + t2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -3t + -0.625 + t2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -3t + t2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -3t + t2 = 0 + -0.625 -3t + t2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -3t + t2 = -0.625 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -0.625 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.625 + 2.25 Combine like terms: -0.625 + 2.25 = 1.625 2.25 + -3t + t2 = 1.625 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1.625 Calculate the square root of the right side: 1.274754878 Break this problem into two subproblems by setting (t + -1.5) equal to 1.274754878 and -1.274754878.Subproblem 1
t + -1.5 = 1.274754878 Simplifying t + -1.5 = 1.274754878 Reorder the terms: -1.5 + t = 1.274754878 Solving -1.5 + t = 1.274754878 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.274754878 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.274754878 + 1.5 t = 1.274754878 + 1.5 Combine like terms: 1.274754878 + 1.5 = 2.774754878 t = 2.774754878 Simplifying t = 2.774754878Subproblem 2
t + -1.5 = -1.274754878 Simplifying t + -1.5 = -1.274754878 Reorder the terms: -1.5 + t = -1.274754878 Solving -1.5 + t = -1.274754878 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.274754878 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.274754878 + 1.5 t = -1.274754878 + 1.5 Combine like terms: -1.274754878 + 1.5 = 0.225245122 t = 0.225245122 Simplifying t = 0.225245122Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.774754878, 0.225245122}
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