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Simplifying 0 = 16t2 + 65t + -9 Reorder the terms: 0 = -9 + 65t + 16t2 Solving 0 = -9 + 65t + 16t2 Solving for variable 't'. Combine like terms: 0 + 9 = 9 9 + -65t + -16t2 = -9 + 65t + 16t2 + 9 + -65t + -16t2 Reorder the terms: 9 + -65t + -16t2 = -9 + 9 + 65t + -65t + 16t2 + -16t2 Combine like terms: -9 + 9 = 0 9 + -65t + -16t2 = 0 + 65t + -65t + 16t2 + -16t2 9 + -65t + -16t2 = 65t + -65t + 16t2 + -16t2 Combine like terms: 65t + -65t = 0 9 + -65t + -16t2 = 0 + 16t2 + -16t2 9 + -65t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 9 + -65t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5625 + 4.0625t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + 4.0625t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + 4.0625t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + 4.0625t + t2 = 0 + 0.5625 4.0625t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 4.0625t + t2 = 0.5625 The t term is 4.0625t. Take half its coefficient (2.03125). Square it (4.125976563) and add it to both sides. Add '4.125976563' to each side of the equation. 4.0625t + 4.125976563 + t2 = 0.5625 + 4.125976563 Reorder the terms: 4.125976563 + 4.0625t + t2 = 0.5625 + 4.125976563 Combine like terms: 0.5625 + 4.125976563 = 4.688476563 4.125976563 + 4.0625t + t2 = 4.688476563 Factor a perfect square on the left side: (t + 2.03125)(t + 2.03125) = 4.688476563 Calculate the square root of the right side: 2.165289025 Break this problem into two subproblems by setting (t + 2.03125) equal to 2.165289025 and -2.165289025.Subproblem 1
t + 2.03125 = 2.165289025 Simplifying t + 2.03125 = 2.165289025 Reorder the terms: 2.03125 + t = 2.165289025 Solving 2.03125 + t = 2.165289025 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.03125' to each side of the equation. 2.03125 + -2.03125 + t = 2.165289025 + -2.03125 Combine like terms: 2.03125 + -2.03125 = 0.00000 0.00000 + t = 2.165289025 + -2.03125 t = 2.165289025 + -2.03125 Combine like terms: 2.165289025 + -2.03125 = 0.134039025 t = 0.134039025 Simplifying t = 0.134039025Subproblem 2
t + 2.03125 = -2.165289025 Simplifying t + 2.03125 = -2.165289025 Reorder the terms: 2.03125 + t = -2.165289025 Solving 2.03125 + t = -2.165289025 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.03125' to each side of the equation. 2.03125 + -2.03125 + t = -2.165289025 + -2.03125 Combine like terms: 2.03125 + -2.03125 = 0.00000 0.00000 + t = -2.165289025 + -2.03125 t = -2.165289025 + -2.03125 Combine like terms: -2.165289025 + -2.03125 = -4.196539025 t = -4.196539025 Simplifying t = -4.196539025Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.134039025, -4.196539025}
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