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Simplifying -2t2 + 12t + 9 = 14.5 Reorder the terms: 9 + 12t + -2t2 = 14.5 Solving 9 + 12t + -2t2 = 14.5 Solving for variable 't'. Reorder the terms: 9 + -14.5 + 12t + -2t2 = 14.5 + -14.5 Combine like terms: 9 + -14.5 = -5.5 -5.5 + 12t + -2t2 = 14.5 + -14.5 Combine like terms: 14.5 + -14.5 = 0.0 -5.5 + 12t + -2t2 = 0.0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 2.75 + -6t + t2 = 0.0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + -6t + -2.75 + t2 = 0.0 + -2.75 Reorder the terms: 2.75 + -2.75 + -6t + t2 = 0.0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + -6t + t2 = 0.0 + -2.75 -6t + t2 = 0.0 + -2.75 Combine like terms: 0.0 + -2.75 = -2.75 -6t + t2 = -2.75 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -2.75 + 9 Reorder the terms: 9 + -6t + t2 = -2.75 + 9 Combine like terms: -2.75 + 9 = 6.25 9 + -6t + t2 = 6.25 Factor a perfect square on the left side: (t + -3)(t + -3) = 6.25 Calculate the square root of the right side: 2.5 Break this problem into two subproblems by setting (t + -3) equal to 2.5 and -2.5.Subproblem 1
t + -3 = 2.5 Simplifying t + -3 = 2.5 Reorder the terms: -3 + t = 2.5 Solving -3 + t = 2.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 2.5 + 3 Combine like terms: -3 + 3 = 0 0 + t = 2.5 + 3 t = 2.5 + 3 Combine like terms: 2.5 + 3 = 5.5 t = 5.5 Simplifying t = 5.5Subproblem 2
t + -3 = -2.5 Simplifying t + -3 = -2.5 Reorder the terms: -3 + t = -2.5 Solving -3 + t = -2.5 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -2.5 + 3 Combine like terms: -3 + 3 = 0 0 + t = -2.5 + 3 t = -2.5 + 3 Combine like terms: -2.5 + 3 = 0.5 t = 0.5 Simplifying t = 0.5Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.5, 0.5}
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