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Simplifying t = -16t2 + 30t + 5 Reorder the terms: t = 5 + 30t + -16t2 Solving t = 5 + 30t + -16t2 Solving for variable 't'. Reorder the terms: -5 + t + -30t + 16t2 = 5 + 30t + -16t2 + -5 + -30t + 16t2 Combine like terms: t + -30t = -29t -5 + -29t + 16t2 = 5 + 30t + -16t2 + -5 + -30t + 16t2 Reorder the terms: -5 + -29t + 16t2 = 5 + -5 + 30t + -30t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -29t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -5 + -29t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -5 + -29t + 16t2 = 0 + -16t2 + 16t2 -5 + -29t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -29t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.8125t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.8125t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.8125t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.8125t + t2 = 0 + 0.3125 -1.8125t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.8125t + t2 = 0.3125 The t term is -1.8125t. Take half its coefficient (-0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. -1.8125t + 0.8212890625 + t2 = 0.3125 + 0.8212890625 Reorder the terms: 0.8212890625 + -1.8125t + t2 = 0.3125 + 0.8212890625 Combine like terms: 0.3125 + 0.8212890625 = 1.1337890625 0.8212890625 + -1.8125t + t2 = 1.1337890625 Factor a perfect square on the left side: (t + -0.90625)(t + -0.90625) = 1.1337890625 Calculate the square root of the right side: 1.064795315 Break this problem into two subproblems by setting (t + -0.90625) equal to 1.064795315 and -1.064795315.Subproblem 1
t + -0.90625 = 1.064795315 Simplifying t + -0.90625 = 1.064795315 Reorder the terms: -0.90625 + t = 1.064795315 Solving -0.90625 + t = 1.064795315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = 1.064795315 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = 1.064795315 + 0.90625 t = 1.064795315 + 0.90625 Combine like terms: 1.064795315 + 0.90625 = 1.971045315 t = 1.971045315 Simplifying t = 1.971045315Subproblem 2
t + -0.90625 = -1.064795315 Simplifying t + -0.90625 = -1.064795315 Reorder the terms: -0.90625 + t = -1.064795315 Solving -0.90625 + t = -1.064795315 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = -1.064795315 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = -1.064795315 + 0.90625 t = -1.064795315 + 0.90625 Combine like terms: -1.064795315 + 0.90625 = -0.158545315 t = -0.158545315 Simplifying t = -0.158545315Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.971045315, -0.158545315}
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