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Simplifying -16t2 + 96t + 180 = 75 Reorder the terms: 180 + 96t + -16t2 = 75 Solving 180 + 96t + -16t2 = 75 Solving for variable 't'. Reorder the terms: 180 + -75 + 96t + -16t2 = 75 + -75 Combine like terms: 180 + -75 = 105 105 + 96t + -16t2 = 75 + -75 Combine like terms: 75 + -75 = 0 105 + 96t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -6.5625 + -6t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + -6t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + -6t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + -6t + t2 = 0 + 6.5625 -6t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 -6t + t2 = 6.5625 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 = 6.5625 + 9 Reorder the terms: 9 + -6t + t2 = 6.5625 + 9 Combine like terms: 6.5625 + 9 = 15.5625 9 + -6t + t2 = 15.5625 Factor a perfect square on the left side: (t + -3)(t + -3) = 15.5625 Calculate the square root of the right side: 3.94493346 Break this problem into two subproblems by setting (t + -3) equal to 3.94493346 and -3.94493346.Subproblem 1
t + -3 = 3.94493346 Simplifying t + -3 = 3.94493346 Reorder the terms: -3 + t = 3.94493346 Solving -3 + t = 3.94493346 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 = 3.94493346 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.94493346 + 3 t = 3.94493346 + 3 Combine like terms: 3.94493346 + 3 = 6.94493346 t = 6.94493346 Simplifying t = 6.94493346Subproblem 2
t + -3 = -3.94493346 Simplifying t + -3 = -3.94493346 Reorder the terms: -3 + t = -3.94493346 Solving -3 + t = -3.94493346 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 = -3.94493346 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.94493346 + 3 t = -3.94493346 + 3 Combine like terms: -3.94493346 + 3 = -0.94493346 t = -0.94493346 Simplifying t = -0.94493346Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.94493346, -0.94493346}
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