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Simplifying 16t2 + 31t + 14 = 0 Reorder the terms: 14 + 31t + 16t2 = 0 Solving 14 + 31t + 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'. 0.875 + 1.9375t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + 1.9375t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + 1.9375t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + 1.9375t + t2 = 0 + -0.875 1.9375t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 1.9375t + t2 = -0.875 The t term is 1.9375t. Take half its coefficient (0.96875). Square it (0.9384765625) and add it to both sides. Add '0.9384765625' to each side of the equation. 1.9375t + 0.9384765625 + t2 = -0.875 + 0.9384765625 Reorder the terms: 0.9384765625 + 1.9375t + t2 = -0.875 + 0.9384765625 Combine like terms: -0.875 + 0.9384765625 = 0.0634765625 0.9384765625 + 1.9375t + t2 = 0.0634765625 Factor a perfect square on the left side: (t + 0.96875)(t + 0.96875) = 0.0634765625 Calculate the square root of the right side: 0.251945555 Break this problem into two subproblems by setting (t + 0.96875) equal to 0.251945555 and -0.251945555.Subproblem 1
t + 0.96875 = 0.251945555 Simplifying t + 0.96875 = 0.251945555 Reorder the terms: 0.96875 + t = 0.251945555 Solving 0.96875 + t = 0.251945555 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.96875' to each side of the equation. 0.96875 + -0.96875 + t = 0.251945555 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + t = 0.251945555 + -0.96875 t = 0.251945555 + -0.96875 Combine like terms: 0.251945555 + -0.96875 = -0.716804445 t = -0.716804445 Simplifying t = -0.716804445Subproblem 2
t + 0.96875 = -0.251945555 Simplifying t + 0.96875 = -0.251945555 Reorder the terms: 0.96875 + t = -0.251945555 Solving 0.96875 + t = -0.251945555 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.96875' to each side of the equation. 0.96875 + -0.96875 + t = -0.251945555 + -0.96875 Combine like terms: 0.96875 + -0.96875 = 0.00000 0.00000 + t = -0.251945555 + -0.96875 t = -0.251945555 + -0.96875 Combine like terms: -0.251945555 + -0.96875 = -1.220695555 t = -1.220695555 Simplifying t = -1.220695555Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.716804445, -1.220695555}
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