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Simplifying 0 = -32t2 + 8t + 3 Reorder the terms: 0 = 3 + 8t + -32t2 Solving 0 = 3 + 8t + -32t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + -8t + 32t2 = 3 + 8t + -32t2 + -3 + -8t + 32t2 Reorder the terms: -3 + -8t + 32t2 = 3 + -3 + 8t + -8t + -32t2 + 32t2 Combine like terms: 3 + -3 = 0 -3 + -8t + 32t2 = 0 + 8t + -8t + -32t2 + 32t2 -3 + -8t + 32t2 = 8t + -8t + -32t2 + 32t2 Combine like terms: 8t + -8t = 0 -3 + -8t + 32t2 = 0 + -32t2 + 32t2 -3 + -8t + 32t2 = -32t2 + 32t2 Combine like terms: -32t2 + 32t2 = 0 -3 + -8t + 32t2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.09375 + -0.25t + t2 = 0 Move the constant term to the right: Add '0.09375' to each side of the equation. -0.09375 + -0.25t + 0.09375 + t2 = 0 + 0.09375 Reorder the terms: -0.09375 + 0.09375 + -0.25t + t2 = 0 + 0.09375 Combine like terms: -0.09375 + 0.09375 = 0.00000 0.00000 + -0.25t + t2 = 0 + 0.09375 -0.25t + t2 = 0 + 0.09375 Combine like terms: 0 + 0.09375 = 0.09375 -0.25t + t2 = 0.09375 The t term is -0.25t. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25t + 0.015625 + t2 = 0.09375 + 0.015625 Reorder the terms: 0.015625 + -0.25t + t2 = 0.09375 + 0.015625 Combine like terms: 0.09375 + 0.015625 = 0.109375 0.015625 + -0.25t + t2 = 0.109375 Factor a perfect square on the left side: (t + -0.125)(t + -0.125) = 0.109375 Calculate the square root of the right side: 0.330718914 Break this problem into two subproblems by setting (t + -0.125) equal to 0.330718914 and -0.330718914.Subproblem 1
t + -0.125 = 0.330718914 Simplifying t + -0.125 = 0.330718914 Reorder the terms: -0.125 + t = 0.330718914 Solving -0.125 + t = 0.330718914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = 0.330718914 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = 0.330718914 + 0.125 t = 0.330718914 + 0.125 Combine like terms: 0.330718914 + 0.125 = 0.455718914 t = 0.455718914 Simplifying t = 0.455718914Subproblem 2
t + -0.125 = -0.330718914 Simplifying t + -0.125 = -0.330718914 Reorder the terms: -0.125 + t = -0.330718914 Solving -0.125 + t = -0.330718914 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + t = -0.330718914 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + t = -0.330718914 + 0.125 t = -0.330718914 + 0.125 Combine like terms: -0.330718914 + 0.125 = -0.205718914 t = -0.205718914 Simplifying t = -0.205718914Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.455718914, -0.205718914}
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