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Simplifying 5 = 6t + -1.5t2 Solving 5 = 6t + -1.5t2 Solving for variable 't'. Reorder the terms: 5 + -6t + 1.5t2 = 6t + -6t + -1.5t2 + 1.5t2 Combine like terms: 6t + -6t = 0 5 + -6t + 1.5t2 = 0 + -1.5t2 + 1.5t2 5 + -6t + 1.5t2 = -1.5t2 + 1.5t2 Combine like terms: -1.5t2 + 1.5t2 = 0.0 5 + -6t + 1.5t2 = 0.0 Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. 3.333333333 + -4t + t2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -4t + -3.333333333 + t2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -4t + t2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -4t + t2 = 0 + -3.333333333 -4t + t2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -4t + t2 = -3.333333333 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -3.333333333 + 4 Reorder the terms: 4 + -4t + t2 = -3.333333333 + 4 Combine like terms: -3.333333333 + 4 = 0.666666667 4 + -4t + t2 = 0.666666667 Factor a perfect square on the left side: (t + -2)(t + -2) = 0.666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (t + -2) equal to 0.816496581 and -0.816496581.Subproblem 1
t + -2 = 0.816496581 Simplifying t + -2 = 0.816496581 Reorder the terms: -2 + t = 0.816496581 Solving -2 + t = 0.816496581 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 0.816496581 + 2 Combine like terms: -2 + 2 = 0 0 + t = 0.816496581 + 2 t = 0.816496581 + 2 Combine like terms: 0.816496581 + 2 = 2.816496581 t = 2.816496581 Simplifying t = 2.816496581Subproblem 2
t + -2 = -0.816496581 Simplifying t + -2 = -0.816496581 Reorder the terms: -2 + t = -0.816496581 Solving -2 + t = -0.816496581 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -0.816496581 + 2 Combine like terms: -2 + 2 = 0 0 + t = -0.816496581 + 2 t = -0.816496581 + 2 Combine like terms: -0.816496581 + 2 = 1.183503419 t = 1.183503419 Simplifying t = 1.183503419Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.816496581, 1.183503419}
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