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Simplifying -16t2 + 30t + -11.5 = 0 Reorder the terms: -11.5 + 30t + -16t2 = 0 Solving -11.5 + 30t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.71875 + -1.875t + t2 = 0 Move the constant term to the right: Add '-0.71875' to each side of the equation. 0.71875 + -1.875t + -0.71875 + t2 = 0 + -0.71875 Reorder the terms: 0.71875 + -0.71875 + -1.875t + t2 = 0 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + -1.875t + t2 = 0 + -0.71875 -1.875t + t2 = 0 + -0.71875 Combine like terms: 0 + -0.71875 = -0.71875 -1.875t + t2 = -0.71875 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 = -0.71875 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = -0.71875 + 0.87890625 Combine like terms: -0.71875 + 0.87890625 = 0.16015625 0.87890625 + -1.875t + t2 = 0.16015625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 0.16015625 Calculate the square root of the right side: 0.400195265 Break this problem into two subproblems by setting (t + -0.9375) equal to 0.400195265 and -0.400195265.Subproblem 1
t + -0.9375 = 0.400195265 Simplifying t + -0.9375 = 0.400195265 Reorder the terms: -0.9375 + t = 0.400195265 Solving -0.9375 + t = 0.400195265 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 = 0.400195265 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 0.400195265 + 0.9375 t = 0.400195265 + 0.9375 Combine like terms: 0.400195265 + 0.9375 = 1.337695265 t = 1.337695265 Simplifying t = 1.337695265Subproblem 2
t + -0.9375 = -0.400195265 Simplifying t + -0.9375 = -0.400195265 Reorder the terms: -0.9375 + t = -0.400195265 Solving -0.9375 + t = -0.400195265 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 = -0.400195265 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -0.400195265 + 0.9375 t = -0.400195265 + 0.9375 Combine like terms: -0.400195265 + 0.9375 = 0.537304735 t = 0.537304735 Simplifying t = 0.537304735Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.337695265, 0.537304735}
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