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Simplifying 0(t) = -16t2 + 32t + 100 Anything times zero is zero. 0t = -16t2 + 32t + 100 Reorder the terms: 0 = 100 + 32t + -16t2 Solving 0 = 100 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0 + -100 = -100 -100 + -32t + 16t2 = 100 + 32t + -16t2 + -100 + -32t + 16t2 Reorder the terms: -100 + -32t + 16t2 = 100 + -100 + 32t + -32t + -16t2 + 16t2 Combine like terms: 100 + -100 = 0 -100 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -100 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -100 + -32t + 16t2 = 0 + -16t2 + 16t2 -100 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -100 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-25 + -8t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-25 + -8t + 4t2)' equal to zero and attempt to solve: Simplifying -25 + -8t + 4t2 = 0 Solving -25 + -8t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.25 + -2t + t2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + -2t + 6.25 + t2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + -2t + t2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + -2t + t2 = 0 + 6.25 -2t + t2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 -2t + t2 = 6.25 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 6.25 + 1 Reorder the terms: 1 + -2t + t2 = 6.25 + 1 Combine like terms: 6.25 + 1 = 7.25 1 + -2t + t2 = 7.25 Factor a perfect square on the left side: (t + -1)(t + -1) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + -1) equal to 2.692582404 and -2.692582404.Subproblem 1
t + -1 = 2.692582404 Simplifying t + -1 = 2.692582404 Reorder the terms: -1 + t = 2.692582404 Solving -1 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 2.692582404 + 1 Combine like terms: -1 + 1 = 0 0 + t = 2.692582404 + 1 t = 2.692582404 + 1 Combine like terms: 2.692582404 + 1 = 3.692582404 t = 3.692582404 Simplifying t = 3.692582404Subproblem 2
t + -1 = -2.692582404 Simplifying t + -1 = -2.692582404 Reorder the terms: -1 + t = -2.692582404 Solving -1 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -2.692582404 + 1 Combine like terms: -1 + 1 = 0 0 + t = -2.692582404 + 1 t = -2.692582404 + 1 Combine like terms: -2.692582404 + 1 = -1.692582404 t = -1.692582404 Simplifying t = -1.692582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.692582404, -1.692582404}Solution
t = {3.692582404, -1.692582404}
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