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Simplifying -4t(t + -2) + 6t = 8t + -3 Reorder the terms: -4t(-2 + t) + 6t = 8t + -3 (-2 * -4t + t * -4t) + 6t = 8t + -3 (8t + -4t2) + 6t = 8t + -3 Reorder the terms: 8t + 6t + -4t2 = 8t + -3 Combine like terms: 8t + 6t = 14t 14t + -4t2 = 8t + -3 Reorder the terms: 14t + -4t2 = -3 + 8t Solving 14t + -4t2 = -3 + 8t Solving for variable 't'. Reorder the terms: 3 + 14t + -8t + -4t2 = -3 + 8t + 3 + -8t Combine like terms: 14t + -8t = 6t 3 + 6t + -4t2 = -3 + 8t + 3 + -8t Reorder the terms: 3 + 6t + -4t2 = -3 + 3 + 8t + -8t Combine like terms: -3 + 3 = 0 3 + 6t + -4t2 = 0 + 8t + -8t 3 + 6t + -4t2 = 8t + -8t Combine like terms: 8t + -8t = 0 3 + 6t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.5t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.5t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.5t + t2 = 0 + 0.75 -1.5t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.5t + t2 = 0.75 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + -1.5t + t2 = 1.3125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (t + -0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
t + -0.75 = 1.145643924 Simplifying t + -0.75 = 1.145643924 Reorder the terms: -0.75 + t = 1.145643924 Solving -0.75 + t = 1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.145643924 + 0.75 t = 1.145643924 + 0.75 Combine like terms: 1.145643924 + 0.75 = 1.895643924 t = 1.895643924 Simplifying t = 1.895643924Subproblem 2
t + -0.75 = -1.145643924 Simplifying t + -0.75 = -1.145643924 Reorder the terms: -0.75 + t = -1.145643924 Solving -0.75 + t = -1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.145643924 + 0.75 t = -1.145643924 + 0.75 Combine like terms: -1.145643924 + 0.75 = -0.395643924 t = -0.395643924 Simplifying t = -0.395643924Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.895643924, -0.395643924}
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