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Simplifying 0 = -16t2 + 65t + 3 Reorder the terms: 0 = 3 + 65t + -16t2 Solving 0 = 3 + 65t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + -65t + 16t2 = 3 + 65t + -16t2 + -3 + -65t + 16t2 Reorder the terms: -3 + -65t + 16t2 = 3 + -3 + 65t + -65t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 -3 + -65t + 16t2 = 0 + 65t + -65t + -16t2 + 16t2 -3 + -65t + 16t2 = 65t + -65t + -16t2 + 16t2 Combine like terms: 65t + -65t = 0 -3 + -65t + 16t2 = 0 + -16t2 + 16t2 -3 + -65t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -65t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -4.0625t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -4.0625t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -4.0625t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -4.0625t + t2 = 0 + 0.1875 -4.0625t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -4.0625t + t2 = 0.1875 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 = 0.1875 + 4.125976563 Reorder the terms: 4.125976563 + -4.0625t + t2 = 0.1875 + 4.125976563 Combine like terms: 0.1875 + 4.125976563 = 4.313476563 4.125976563 + -4.0625t + t2 = 4.313476563 Factor a perfect square on the left side: (t + -2.03125)(t + -2.03125) = 4.313476563 Calculate the square root of the right side: 2.076891081 Break this problem into two subproblems by setting (t + -2.03125) equal to 2.076891081 and -2.076891081.Subproblem 1
t + -2.03125 = 2.076891081 Simplifying t + -2.03125 = 2.076891081 Reorder the terms: -2.03125 + t = 2.076891081 Solving -2.03125 + t = 2.076891081 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 = 2.076891081 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = 2.076891081 + 2.03125 t = 2.076891081 + 2.03125 Combine like terms: 2.076891081 + 2.03125 = 4.108141081 t = 4.108141081 Simplifying t = 4.108141081Subproblem 2
t + -2.03125 = -2.076891081 Simplifying t + -2.03125 = -2.076891081 Reorder the terms: -2.03125 + t = -2.076891081 Solving -2.03125 + t = -2.076891081 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 = -2.076891081 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = -2.076891081 + 2.03125 t = -2.076891081 + 2.03125 Combine like terms: -2.076891081 + 2.03125 = -0.045641081 t = -0.045641081 Simplifying t = -0.045641081Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.108141081, -0.045641081}
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