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Simplifying -4t(t + -1) + 6t = 4t + -9 Reorder the terms: -4t(-1 + t) + 6t = 4t + -9 (-1 * -4t + t * -4t) + 6t = 4t + -9 (4t + -4t2) + 6t = 4t + -9 Reorder the terms: 4t + 6t + -4t2 = 4t + -9 Combine like terms: 4t + 6t = 10t 10t + -4t2 = 4t + -9 Reorder the terms: 10t + -4t2 = -9 + 4t Solving 10t + -4t2 = -9 + 4t Solving for variable 't'. Reorder the terms: 9 + 10t + -4t + -4t2 = -9 + 4t + 9 + -4t Combine like terms: 10t + -4t = 6t 9 + 6t + -4t2 = -9 + 4t + 9 + -4t Reorder the terms: 9 + 6t + -4t2 = -9 + 9 + 4t + -4t Combine like terms: -9 + 9 = 0 9 + 6t + -4t2 = 0 + 4t + -4t 9 + 6t + -4t2 = 4t + -4t Combine like terms: 4t + -4t = 0 9 + 6t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + -1.5t + t2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -1.5t + 2.25 + t2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -1.5t + t2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -1.5t + t2 = 0 + 2.25 -1.5t + t2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -1.5t + t2 = 2.25 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 = 2.25 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 2.25 + 0.5625 Combine like terms: 2.25 + 0.5625 = 2.8125 0.5625 + -1.5t + t2 = 2.8125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (t + -0.75) equal to 1.677050983 and -1.677050983.Subproblem 1
t + -0.75 = 1.677050983 Simplifying t + -0.75 = 1.677050983 Reorder the terms: -0.75 + t = 1.677050983 Solving -0.75 + t = 1.677050983 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 = 1.677050983 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.677050983 + 0.75 t = 1.677050983 + 0.75 Combine like terms: 1.677050983 + 0.75 = 2.427050983 t = 2.427050983 Simplifying t = 2.427050983Subproblem 2
t + -0.75 = -1.677050983 Simplifying t + -0.75 = -1.677050983 Reorder the terms: -0.75 + t = -1.677050983 Solving -0.75 + t = -1.677050983 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 = -1.677050983 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.677050983 + 0.75 t = -1.677050983 + 0.75 Combine like terms: -1.677050983 + 0.75 = -0.927050983 t = -0.927050983 Simplifying t = -0.927050983Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.427050983, -0.927050983}
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