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Simplifying -2(t + -4) * 4t = 5t + -3 Reorder the terms: -2(-4 + t) * 4t = 5t + -3 Reorder the terms for easier multiplication: -2 * 4t(-4 + t) = 5t + -3 Multiply -2 * 4 -8t(-4 + t) = 5t + -3 (-4 * -8t + t * -8t) = 5t + -3 (32t + -8t2) = 5t + -3 Reorder the terms: 32t + -8t2 = -3 + 5t Solving 32t + -8t2 = -3 + 5t Solving for variable 't'. Reorder the terms: 3 + 32t + -5t + -8t2 = -3 + 5t + 3 + -5t Combine like terms: 32t + -5t = 27t 3 + 27t + -8t2 = -3 + 5t + 3 + -5t Reorder the terms: 3 + 27t + -8t2 = -3 + 3 + 5t + -5t Combine like terms: -3 + 3 = 0 3 + 27t + -8t2 = 0 + 5t + -5t 3 + 27t + -8t2 = 5t + -5t Combine like terms: 5t + -5t = 0 3 + 27t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.375 + -3.375t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -3.375t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -3.375t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -3.375t + t2 = 0 + 0.375 -3.375t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -3.375t + t2 = 0.375 The t term is -3.375t. Take half its coefficient (-1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. -3.375t + 2.84765625 + t2 = 0.375 + 2.84765625 Reorder the terms: 2.84765625 + -3.375t + t2 = 0.375 + 2.84765625 Combine like terms: 0.375 + 2.84765625 = 3.22265625 2.84765625 + -3.375t + t2 = 3.22265625 Factor a perfect square on the left side: (t + -1.6875)(t + -1.6875) = 3.22265625 Calculate the square root of the right side: 1.795175827 Break this problem into two subproblems by setting (t + -1.6875) equal to 1.795175827 and -1.795175827.Subproblem 1
t + -1.6875 = 1.795175827 Simplifying t + -1.6875 = 1.795175827 Reorder the terms: -1.6875 + t = 1.795175827 Solving -1.6875 + t = 1.795175827 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + t = 1.795175827 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + t = 1.795175827 + 1.6875 t = 1.795175827 + 1.6875 Combine like terms: 1.795175827 + 1.6875 = 3.482675827 t = 3.482675827 Simplifying t = 3.482675827Subproblem 2
t + -1.6875 = -1.795175827 Simplifying t + -1.6875 = -1.795175827 Reorder the terms: -1.6875 + t = -1.795175827 Solving -1.6875 + t = -1.795175827 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + t = -1.795175827 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + t = -1.795175827 + 1.6875 t = -1.795175827 + 1.6875 Combine like terms: -1.795175827 + 1.6875 = -0.107675827 t = -0.107675827 Simplifying t = -0.107675827Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.482675827, -0.107675827}
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