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Simplifying 0 = -16t2 + 36t + 9 Reorder the terms: 0 = 9 + 36t + -16t2 Solving 0 = 9 + 36t + -16t2 Solving for variable 't'. Combine like terms: 0 + -9 = -9 -9 + -36t + 16t2 = 9 + 36t + -16t2 + -9 + -36t + 16t2 Reorder the terms: -9 + -36t + 16t2 = 9 + -9 + 36t + -36t + -16t2 + 16t2 Combine like terms: 9 + -9 = 0 -9 + -36t + 16t2 = 0 + 36t + -36t + -16t2 + 16t2 -9 + -36t + 16t2 = 36t + -36t + -16t2 + 16t2 Combine like terms: 36t + -36t = 0 -9 + -36t + 16t2 = 0 + -16t2 + 16t2 -9 + -36t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -9 + -36t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.5625 + -2.25t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -2.25t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -2.25t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -2.25t + t2 = 0 + 0.5625 -2.25t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -2.25t + t2 = 0.5625 The t term is -2.25t. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25t + 1.265625 + t2 = 0.5625 + 1.265625 Reorder the terms: 1.265625 + -2.25t + t2 = 0.5625 + 1.265625 Combine like terms: 0.5625 + 1.265625 = 1.828125 1.265625 + -2.25t + t2 = 1.828125 Factor a perfect square on the left side: (t + -1.125)(t + -1.125) = 1.828125 Calculate the square root of the right side: 1.352081728 Break this problem into two subproblems by setting (t + -1.125) equal to 1.352081728 and -1.352081728.Subproblem 1
t + -1.125 = 1.352081728 Simplifying t + -1.125 = 1.352081728 Reorder the terms: -1.125 + t = 1.352081728 Solving -1.125 + t = 1.352081728 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = 1.352081728 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = 1.352081728 + 1.125 t = 1.352081728 + 1.125 Combine like terms: 1.352081728 + 1.125 = 2.477081728 t = 2.477081728 Simplifying t = 2.477081728Subproblem 2
t + -1.125 = -1.352081728 Simplifying t + -1.125 = -1.352081728 Reorder the terms: -1.125 + t = -1.352081728 Solving -1.125 + t = -1.352081728 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + t = -1.352081728 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + t = -1.352081728 + 1.125 t = -1.352081728 + 1.125 Combine like terms: -1.352081728 + 1.125 = -0.227081728 t = -0.227081728 Simplifying t = -0.227081728Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.477081728, -0.227081728}
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