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Simplifying 0 = -16t2 + 13t + 16 Reorder the terms: 0 = 16 + 13t + -16t2 Solving 0 = 16 + 13t + -16t2 Solving for variable 't'. Combine like terms: 0 + -16 = -16 -16 + -13t + 16t2 = 16 + 13t + -16t2 + -16 + -13t + 16t2 Reorder the terms: -16 + -13t + 16t2 = 16 + -16 + 13t + -13t + -16t2 + 16t2 Combine like terms: 16 + -16 = 0 -16 + -13t + 16t2 = 0 + 13t + -13t + -16t2 + 16t2 -16 + -13t + 16t2 = 13t + -13t + -16t2 + 16t2 Combine like terms: 13t + -13t = 0 -16 + -13t + 16t2 = 0 + -16t2 + 16t2 -16 + -13t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -16 + -13t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1 + -0.8125t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.8125t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -0.8125t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.8125t + t2 = 0 + 1 -0.8125t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.8125t + t2 = 1 The t term is -0.8125t. Take half its coefficient (-0.40625). Square it (0.1650390625) and add it to both sides. Add '0.1650390625' to each side of the equation. -0.8125t + 0.1650390625 + t2 = 1 + 0.1650390625 Reorder the terms: 0.1650390625 + -0.8125t + t2 = 1 + 0.1650390625 Combine like terms: 1 + 0.1650390625 = 1.1650390625 0.1650390625 + -0.8125t + t2 = 1.1650390625 Factor a perfect square on the left side: (t + -0.40625)(t + -0.40625) = 1.1650390625 Calculate the square root of the right side: 1.079369752 Break this problem into two subproblems by setting (t + -0.40625) equal to 1.079369752 and -1.079369752.Subproblem 1
t + -0.40625 = 1.079369752 Simplifying t + -0.40625 = 1.079369752 Reorder the terms: -0.40625 + t = 1.079369752 Solving -0.40625 + t = 1.079369752 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.40625' to each side of the equation. -0.40625 + 0.40625 + t = 1.079369752 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + t = 1.079369752 + 0.40625 t = 1.079369752 + 0.40625 Combine like terms: 1.079369752 + 0.40625 = 1.485619752 t = 1.485619752 Simplifying t = 1.485619752Subproblem 2
t + -0.40625 = -1.079369752 Simplifying t + -0.40625 = -1.079369752 Reorder the terms: -0.40625 + t = -1.079369752 Solving -0.40625 + t = -1.079369752 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.40625' to each side of the equation. -0.40625 + 0.40625 + t = -1.079369752 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + t = -1.079369752 + 0.40625 t = -1.079369752 + 0.40625 Combine like terms: -1.079369752 + 0.40625 = -0.673119752 t = -0.673119752 Simplifying t = -0.673119752Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.485619752, -0.673119752}
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