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Simplifying 1 = -16t2 + 30t + 75 Reorder the terms: 1 = 75 + 30t + -16t2 Solving 1 = 75 + 30t + -16t2 Solving for variable 't'. Combine like terms: 1 + -75 = -74 -74 + -30t + 16t2 = 75 + 30t + -16t2 + -75 + -30t + 16t2 Reorder the terms: -74 + -30t + 16t2 = 75 + -75 + 30t + -30t + -16t2 + 16t2 Combine like terms: 75 + -75 = 0 -74 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -74 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -74 + -30t + 16t2 = 0 + -16t2 + 16t2 -74 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -74 + -30t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-37 + -15t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-37 + -15t + 8t2)' equal to zero and attempt to solve: Simplifying -37 + -15t + 8t2 = 0 Solving -37 + -15t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.625 + -1.875t + t2 = 0 Move the constant term to the right: Add '4.625' to each side of the equation. -4.625 + -1.875t + 4.625 + t2 = 0 + 4.625 Reorder the terms: -4.625 + 4.625 + -1.875t + t2 = 0 + 4.625 Combine like terms: -4.625 + 4.625 = 0.000 0.000 + -1.875t + t2 = 0 + 4.625 -1.875t + t2 = 0 + 4.625 Combine like terms: 0 + 4.625 = 4.625 -1.875t + t2 = 4.625 The t term is -1.875t. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875t + 0.87890625 + t2 = 4.625 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 4.625 + 0.87890625 Combine like terms: 4.625 + 0.87890625 = 5.50390625 0.87890625 + -1.875t + t2 = 5.50390625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 5.50390625 Calculate the square root of the right side: 2.346040547 Break this problem into two subproblems by setting (t + -0.9375) equal to 2.346040547 and -2.346040547.Subproblem 1
t + -0.9375 = 2.346040547 Simplifying t + -0.9375 = 2.346040547 Reorder the terms: -0.9375 + t = 2.346040547 Solving -0.9375 + t = 2.346040547 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = 2.346040547 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 2.346040547 + 0.9375 t = 2.346040547 + 0.9375 Combine like terms: 2.346040547 + 0.9375 = 3.283540547 t = 3.283540547 Simplifying t = 3.283540547Subproblem 2
t + -0.9375 = -2.346040547 Simplifying t + -0.9375 = -2.346040547 Reorder the terms: -0.9375 + t = -2.346040547 Solving -0.9375 + t = -2.346040547 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = -2.346040547 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -2.346040547 + 0.9375 t = -2.346040547 + 0.9375 Combine like terms: -2.346040547 + 0.9375 = -1.408540547 t = -1.408540547 Simplifying t = -1.408540547Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.283540547, -1.408540547}Solution
t = {3.283540547, -1.408540547}
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