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Simplifying 0 = -16t2 + 40t + 14 Reorder the terms: 0 = 14 + 40t + -16t2 Solving 0 = 14 + 40t + -16t2 Solving for variable 't'. Combine like terms: 0 + -14 = -14 -14 + -40t + 16t2 = 14 + 40t + -16t2 + -14 + -40t + 16t2 Reorder the terms: -14 + -40t + 16t2 = 14 + -14 + 40t + -40t + -16t2 + 16t2 Combine like terms: 14 + -14 = 0 -14 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 -14 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -14 + -40t + 16t2 = 0 + -16t2 + 16t2 -14 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -14 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -20t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -20t + 8t2)' equal to zero and attempt to solve: Simplifying -7 + -20t + 8t2 = 0 Solving -7 + -20t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + -2.5t + t2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -2.5t + 0.875 + t2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -2.5t + t2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -2.5t + t2 = 0 + 0.875 -2.5t + t2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -2.5t + t2 = 0.875 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 0.875 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 0.875 + 1.5625 Combine like terms: 0.875 + 1.5625 = 2.4375 1.5625 + -2.5t + t2 = 2.4375 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 2.4375 Calculate the square root of the right side: 1.5612495 Break this problem into two subproblems by setting (t + -1.25) equal to 1.5612495 and -1.5612495.Subproblem 1
t + -1.25 = 1.5612495 Simplifying t + -1.25 = 1.5612495 Reorder the terms: -1.25 + t = 1.5612495 Solving -1.25 + t = 1.5612495 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.5612495 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.5612495 + 1.25 t = 1.5612495 + 1.25 Combine like terms: 1.5612495 + 1.25 = 2.8112495 t = 2.8112495 Simplifying t = 2.8112495Subproblem 2
t + -1.25 = -1.5612495 Simplifying t + -1.25 = -1.5612495 Reorder the terms: -1.25 + t = -1.5612495 Solving -1.25 + t = -1.5612495 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.5612495 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.5612495 + 1.25 t = -1.5612495 + 1.25 Combine like terms: -1.5612495 + 1.25 = -0.3112495 t = -0.3112495 Simplifying t = -0.3112495Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.8112495, -0.3112495}Solution
t = {2.8112495, -0.3112495}
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