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Simplifying 12 = 4 + 23t + 16t2 Solving 12 = 4 + 23t + 16t2 Solving for variable 't'. Combine like terms: 12 + -4 = 8 8 + -23t + -16t2 = 4 + 23t + 16t2 + -4 + -23t + -16t2 Reorder the terms: 8 + -23t + -16t2 = 4 + -4 + 23t + -23t + 16t2 + -16t2 Combine like terms: 4 + -4 = 0 8 + -23t + -16t2 = 0 + 23t + -23t + 16t2 + -16t2 8 + -23t + -16t2 = 23t + -23t + 16t2 + -16t2 Combine like terms: 23t + -23t = 0 8 + -23t + -16t2 = 0 + 16t2 + -16t2 8 + -23t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 8 + -23t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5 + 1.4375t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.4375t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.4375t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.4375t + t2 = 0 + 0.5 1.4375t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.4375t + t2 = 0.5 The t term is 1.4375t. Take half its coefficient (0.71875). Square it (0.5166015625) and add it to both sides. Add '0.5166015625' to each side of the equation. 1.4375t + 0.5166015625 + t2 = 0.5 + 0.5166015625 Reorder the terms: 0.5166015625 + 1.4375t + t2 = 0.5 + 0.5166015625 Combine like terms: 0.5 + 0.5166015625 = 1.0166015625 0.5166015625 + 1.4375t + t2 = 1.0166015625 Factor a perfect square on the left side: (t + 0.71875)(t + 0.71875) = 1.0166015625 Calculate the square root of the right side: 1.008266613 Break this problem into two subproblems by setting (t + 0.71875) equal to 1.008266613 and -1.008266613.Subproblem 1
t + 0.71875 = 1.008266613 Simplifying t + 0.71875 = 1.008266613 Reorder the terms: 0.71875 + t = 1.008266613 Solving 0.71875 + t = 1.008266613 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.71875' to each side of the equation. 0.71875 + -0.71875 + t = 1.008266613 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + t = 1.008266613 + -0.71875 t = 1.008266613 + -0.71875 Combine like terms: 1.008266613 + -0.71875 = 0.289516613 t = 0.289516613 Simplifying t = 0.289516613Subproblem 2
t + 0.71875 = -1.008266613 Simplifying t + 0.71875 = -1.008266613 Reorder the terms: 0.71875 + t = -1.008266613 Solving 0.71875 + t = -1.008266613 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.71875' to each side of the equation. 0.71875 + -0.71875 + t = -1.008266613 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + t = -1.008266613 + -0.71875 t = -1.008266613 + -0.71875 Combine like terms: -1.008266613 + -0.71875 = -1.727016613 t = -1.727016613 Simplifying t = -1.727016613Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.289516613, -1.727016613}
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