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Simplifying 21 = 5 + 34t + -16t2 Solving 21 = 5 + 34t + -16t2 Solving for variable 't'. Combine like terms: 21 + -5 = 16 16 + -34t + 16t2 = 5 + 34t + -16t2 + -5 + -34t + 16t2 Reorder the terms: 16 + -34t + 16t2 = 5 + -5 + 34t + -34t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 16 + -34t + 16t2 = 0 + 34t + -34t + -16t2 + 16t2 16 + -34t + 16t2 = 34t + -34t + -16t2 + 16t2 Combine like terms: 34t + -34t = 0 16 + -34t + 16t2 = 0 + -16t2 + 16t2 16 + -34t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 16 + -34t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -17t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -17t + 8t2)' equal to zero and attempt to solve: Simplifying 8 + -17t + 8t2 = 0 Solving 8 + -17t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1 + -2.125t + t2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.125t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -2.125t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.125t + t2 = 0 + -1 -2.125t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.125t + t2 = -1 The t term is -2.125t. Take half its coefficient (-1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. -2.125t + 1.12890625 + t2 = -1 + 1.12890625 Reorder the terms: 1.12890625 + -2.125t + t2 = -1 + 1.12890625 Combine like terms: -1 + 1.12890625 = 0.12890625 1.12890625 + -2.125t + t2 = 0.12890625 Factor a perfect square on the left side: (t + -1.0625)(t + -1.0625) = 0.12890625 Calculate the square root of the right side: 0.359035165 Break this problem into two subproblems by setting (t + -1.0625) equal to 0.359035165 and -0.359035165.Subproblem 1
t + -1.0625 = 0.359035165 Simplifying t + -1.0625 = 0.359035165 Reorder the terms: -1.0625 + t = 0.359035165 Solving -1.0625 + t = 0.359035165 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = 0.359035165 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = 0.359035165 + 1.0625 t = 0.359035165 + 1.0625 Combine like terms: 0.359035165 + 1.0625 = 1.421535165 t = 1.421535165 Simplifying t = 1.421535165Subproblem 2
t + -1.0625 = -0.359035165 Simplifying t + -1.0625 = -0.359035165 Reorder the terms: -1.0625 + t = -0.359035165 Solving -1.0625 + t = -0.359035165 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = -0.359035165 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = -0.359035165 + 1.0625 t = -0.359035165 + 1.0625 Combine like terms: -0.359035165 + 1.0625 = 0.703464835 t = 0.703464835 Simplifying t = 0.703464835Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.421535165, 0.703464835}Solution
t = {1.421535165, 0.703464835}
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