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Simplifying -16t2 + 45t + 6.5 = 7 Reorder the terms: 6.5 + 45t + -16t2 = 7 Solving 6.5 + 45t + -16t2 = 7 Solving for variable 't'. Reorder the terms: 6.5 + -7 + 45t + -16t2 = 7 + -7 Combine like terms: 6.5 + -7 = -0.5 -0.5 + 45t + -16t2 = 7 + -7 Combine like terms: 7 + -7 = 0 -0.5 + 45t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.03125 + -2.8125t + t2 = 0 Move the constant term to the right: Add '-0.03125' to each side of the equation. 0.03125 + -2.8125t + -0.03125 + t2 = 0 + -0.03125 Reorder the terms: 0.03125 + -0.03125 + -2.8125t + t2 = 0 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + -2.8125t + t2 = 0 + -0.03125 -2.8125t + t2 = 0 + -0.03125 Combine like terms: 0 + -0.03125 = -0.03125 -2.8125t + t2 = -0.03125 The t term is -2.8125t. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125t + 1.977539063 + t2 = -0.03125 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = -0.03125 + 1.977539063 Combine like terms: -0.03125 + 1.977539063 = 1.946289063 1.977539063 + -2.8125t + t2 = 1.946289063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 1.946289063 Calculate the square root of the right side: 1.395094643 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.395094643 and -1.395094643.Subproblem 1
t + -1.40625 = 1.395094643 Simplifying t + -1.40625 = 1.395094643 Reorder the terms: -1.40625 + t = 1.395094643 Solving -1.40625 + t = 1.395094643 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = 1.395094643 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.395094643 + 1.40625 t = 1.395094643 + 1.40625 Combine like terms: 1.395094643 + 1.40625 = 2.801344643 t = 2.801344643 Simplifying t = 2.801344643Subproblem 2
t + -1.40625 = -1.395094643 Simplifying t + -1.40625 = -1.395094643 Reorder the terms: -1.40625 + t = -1.395094643 Solving -1.40625 + t = -1.395094643 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = -1.395094643 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.395094643 + 1.40625 t = -1.395094643 + 1.40625 Combine like terms: -1.395094643 + 1.40625 = 0.011155357 t = 0.011155357 Simplifying t = 0.011155357Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.801344643, 0.011155357}
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