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Simplifying 1 = 15t + -5t2 Solving 1 = 15t + -5t2 Solving for variable 't'. Reorder the terms: 1 + -15t + 5t2 = 15t + -15t + -5t2 + 5t2 Combine like terms: 15t + -15t = 0 1 + -15t + 5t2 = 0 + -5t2 + 5t2 1 + -15t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 1 + -15t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + -3t + t2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -3t + -0.2 + t2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -3t + t2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -3t + t2 = 0 + -0.2 -3t + t2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -3t + t2 = -0.2 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 = -0.2 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.2 + 2.25 Combine like terms: -0.2 + 2.25 = 2.05 2.25 + -3t + t2 = 2.05 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 2.05 Calculate the square root of the right side: 1.431782106 Break this problem into two subproblems by setting (t + -1.5) equal to 1.431782106 and -1.431782106.Subproblem 1
t + -1.5 = 1.431782106 Simplifying t + -1.5 = 1.431782106 Reorder the terms: -1.5 + t = 1.431782106 Solving -1.5 + t = 1.431782106 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 = 1.431782106 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.431782106 + 1.5 t = 1.431782106 + 1.5 Combine like terms: 1.431782106 + 1.5 = 2.931782106 t = 2.931782106 Simplifying t = 2.931782106Subproblem 2
t + -1.5 = -1.431782106 Simplifying t + -1.5 = -1.431782106 Reorder the terms: -1.5 + t = -1.431782106 Solving -1.5 + t = -1.431782106 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 = -1.431782106 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.431782106 + 1.5 t = -1.431782106 + 1.5 Combine like terms: -1.431782106 + 1.5 = 0.068217894 t = 0.068217894 Simplifying t = 0.068217894Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.931782106, 0.068217894}
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