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Simplifying 0.5 = -16t2 + 48t + 7 Reorder the terms: 0.5 = 7 + 48t + -16t2 Solving 0.5 = 7 + 48t + -16t2 Solving for variable 't'. Combine like terms: 0.5 + -7 = -6.5 -6.5 + -48t + 16t2 = 7 + 48t + -16t2 + -7 + -48t + 16t2 Reorder the terms: -6.5 + -48t + 16t2 = 7 + -7 + 48t + -48t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 -6.5 + -48t + 16t2 = 0 + 48t + -48t + -16t2 + 16t2 -6.5 + -48t + 16t2 = 48t + -48t + -16t2 + 16t2 Combine like terms: 48t + -48t = 0 -6.5 + -48t + 16t2 = 0 + -16t2 + 16t2 -6.5 + -48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6.5 + -48t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.40625 + -3t + t2 = 0 Move the constant term to the right: Add '0.40625' to each side of the equation. -0.40625 + -3t + 0.40625 + t2 = 0 + 0.40625 Reorder the terms: -0.40625 + 0.40625 + -3t + t2 = 0 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + -3t + t2 = 0 + 0.40625 -3t + t2 = 0 + 0.40625 Combine like terms: 0 + 0.40625 = 0.40625 -3t + t2 = 0.40625 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 0.40625 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 0.40625 + 2.25 Combine like terms: 0.40625 + 2.25 = 2.65625 2.25 + -3t + t2 = 2.65625 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 2.65625 Calculate the square root of the right side: 1.629800601 Break this problem into two subproblems by setting (t + -1.5) equal to 1.629800601 and -1.629800601.Subproblem 1
t + -1.5 = 1.629800601 Simplifying t + -1.5 = 1.629800601 Reorder the terms: -1.5 + t = 1.629800601 Solving -1.5 + t = 1.629800601 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.629800601 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.629800601 + 1.5 t = 1.629800601 + 1.5 Combine like terms: 1.629800601 + 1.5 = 3.129800601 t = 3.129800601 Simplifying t = 3.129800601Subproblem 2
t + -1.5 = -1.629800601 Simplifying t + -1.5 = -1.629800601 Reorder the terms: -1.5 + t = -1.629800601 Solving -1.5 + t = -1.629800601 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.629800601 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.629800601 + 1.5 t = -1.629800601 + 1.5 Combine like terms: -1.629800601 + 1.5 = -0.129800601 t = -0.129800601 Simplifying t = -0.129800601Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.129800601, -0.129800601}
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