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Simplifying -16t2 + 30t + 105 = 0 Reorder the terms: 105 + 30t + -16t2 = 0 Solving 105 + 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'. -6.5625 + -1.875t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + -1.875t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + -1.875t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + -1.875t + t2 = 0 + 6.5625 -1.875t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 -1.875t + t2 = 6.5625 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 = 6.5625 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 6.5625 + 0.87890625 Combine like terms: 6.5625 + 0.87890625 = 7.44140625 0.87890625 + -1.875t + t2 = 7.44140625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 7.44140625 Calculate the square root of the right side: 2.727894105 Break this problem into two subproblems by setting (t + -0.9375) equal to 2.727894105 and -2.727894105.Subproblem 1
t + -0.9375 = 2.727894105 Simplifying t + -0.9375 = 2.727894105 Reorder the terms: -0.9375 + t = 2.727894105 Solving -0.9375 + t = 2.727894105 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 = 2.727894105 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 2.727894105 + 0.9375 t = 2.727894105 + 0.9375 Combine like terms: 2.727894105 + 0.9375 = 3.665394105 t = 3.665394105 Simplifying t = 3.665394105Subproblem 2
t + -0.9375 = -2.727894105 Simplifying t + -0.9375 = -2.727894105 Reorder the terms: -0.9375 + t = -2.727894105 Solving -0.9375 + t = -2.727894105 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 = -2.727894105 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -2.727894105 + 0.9375 t = -2.727894105 + 0.9375 Combine like terms: -2.727894105 + 0.9375 = -1.790394105 t = -1.790394105 Simplifying t = -1.790394105Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.665394105, -1.790394105}
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