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Simplifying -16t2 + 80t + 16 = 0 Reorder the terms: 16 + 80t + -16t2 = 0 Solving 16 + 80t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(1 + 5t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(1 + 5t + -1t2)' equal to zero and attempt to solve: Simplifying 1 + 5t + -1t2 = 0 Solving 1 + 5t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -5t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -5t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -5t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -5t + t2 = 0 + 1 -5t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -5t + t2 = 1 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = 1 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 1 + 6.25 Combine like terms: 1 + 6.25 = 7.25 6.25 + -5t + t2 = 7.25 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + -2.5) equal to 2.692582404 and -2.692582404.Subproblem 1
t + -2.5 = 2.692582404 Simplifying t + -2.5 = 2.692582404 Reorder the terms: -2.5 + t = 2.692582404 Solving -2.5 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.692582404 + 2.5 t = 2.692582404 + 2.5 Combine like terms: 2.692582404 + 2.5 = 5.192582404 t = 5.192582404 Simplifying t = 5.192582404Subproblem 2
t + -2.5 = -2.692582404 Simplifying t + -2.5 = -2.692582404 Reorder the terms: -2.5 + t = -2.692582404 Solving -2.5 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.692582404 + 2.5 t = -2.692582404 + 2.5 Combine like terms: -2.692582404 + 2.5 = -0.192582404 t = -0.192582404 Simplifying t = -0.192582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.192582404, -0.192582404}Solution
t = {5.192582404, -0.192582404}
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