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Simplifying 0 = -16t2 + 113t + 65 Reorder the terms: 0 = 65 + 113t + -16t2 Solving 0 = 65 + 113t + -16t2 Solving for variable 't'. Combine like terms: 0 + -65 = -65 -65 + -113t + 16t2 = 65 + 113t + -16t2 + -65 + -113t + 16t2 Reorder the terms: -65 + -113t + 16t2 = 65 + -65 + 113t + -113t + -16t2 + 16t2 Combine like terms: 65 + -65 = 0 -65 + -113t + 16t2 = 0 + 113t + -113t + -16t2 + 16t2 -65 + -113t + 16t2 = 113t + -113t + -16t2 + 16t2 Combine like terms: 113t + -113t = 0 -65 + -113t + 16t2 = 0 + -16t2 + 16t2 -65 + -113t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -65 + -113t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.0625 + -7.0625t + t2 = 0 Move the constant term to the right: Add '4.0625' to each side of the equation. -4.0625 + -7.0625t + 4.0625 + t2 = 0 + 4.0625 Reorder the terms: -4.0625 + 4.0625 + -7.0625t + t2 = 0 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + -7.0625t + t2 = 0 + 4.0625 -7.0625t + t2 = 0 + 4.0625 Combine like terms: 0 + 4.0625 = 4.0625 -7.0625t + t2 = 4.0625 The t term is -7.0625t. Take half its coefficient (-3.53125). Square it (12.46972656) and add it to both sides. Add '12.46972656' to each side of the equation. -7.0625t + 12.46972656 + t2 = 4.0625 + 12.46972656 Reorder the terms: 12.46972656 + -7.0625t + t2 = 4.0625 + 12.46972656 Combine like terms: 4.0625 + 12.46972656 = 16.53222656 12.46972656 + -7.0625t + t2 = 16.53222656 Factor a perfect square on the left side: (t + -3.53125)(t + -3.53125) = 16.53222656 Calculate the square root of the right side: 4.065984083 Break this problem into two subproblems by setting (t + -3.53125) equal to 4.065984083 and -4.065984083.Subproblem 1
t + -3.53125 = 4.065984083 Simplifying t + -3.53125 = 4.065984083 Reorder the terms: -3.53125 + t = 4.065984083 Solving -3.53125 + t = 4.065984083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.53125' to each side of the equation. -3.53125 + 3.53125 + t = 4.065984083 + 3.53125 Combine like terms: -3.53125 + 3.53125 = 0.00000 0.00000 + t = 4.065984083 + 3.53125 t = 4.065984083 + 3.53125 Combine like terms: 4.065984083 + 3.53125 = 7.597234083 t = 7.597234083 Simplifying t = 7.597234083Subproblem 2
t + -3.53125 = -4.065984083 Simplifying t + -3.53125 = -4.065984083 Reorder the terms: -3.53125 + t = -4.065984083 Solving -3.53125 + t = -4.065984083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.53125' to each side of the equation. -3.53125 + 3.53125 + t = -4.065984083 + 3.53125 Combine like terms: -3.53125 + 3.53125 = 0.00000 0.00000 + t = -4.065984083 + 3.53125 t = -4.065984083 + 3.53125 Combine like terms: -4.065984083 + 3.53125 = -0.534734083 t = -0.534734083 Simplifying t = -0.534734083Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.597234083, -0.534734083}
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