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Simplifying 32 = 16t2 + 72t Reorder the terms: 32 = 72t + 16t2 Solving 32 = 72t + 16t2 Solving for variable 't'. Reorder the terms: 32 + -72t + -16t2 = 72t + -72t + 16t2 + -16t2 Combine like terms: 72t + -72t = 0 32 + -72t + -16t2 = 0 + 16t2 + -16t2 32 + -72t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 32 + -72t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(4 + -9t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(4 + -9t + -2t2)' equal to zero and attempt to solve: Simplifying 4 + -9t + -2t2 = 0 Solving 4 + -9t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + 4.5t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 4.5t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 4.5t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 4.5t + t2 = 0 + 2 4.5t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 4.5t + t2 = 2 The t term is 4.5t. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5t + 5.0625 + t2 = 2 + 5.0625 Reorder the terms: 5.0625 + 4.5t + t2 = 2 + 5.0625 Combine like terms: 2 + 5.0625 = 7.0625 5.0625 + 4.5t + t2 = 7.0625 Factor a perfect square on the left side: (t + 2.25)(t + 2.25) = 7.0625 Calculate the square root of the right side: 2.657536453 Break this problem into two subproblems by setting (t + 2.25) equal to 2.657536453 and -2.657536453.Subproblem 1
t + 2.25 = 2.657536453 Simplifying t + 2.25 = 2.657536453 Reorder the terms: 2.25 + t = 2.657536453 Solving 2.25 + t = 2.657536453 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = 2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = 2.657536453 + -2.25 t = 2.657536453 + -2.25 Combine like terms: 2.657536453 + -2.25 = 0.407536453 t = 0.407536453 Simplifying t = 0.407536453Subproblem 2
t + 2.25 = -2.657536453 Simplifying t + 2.25 = -2.657536453 Reorder the terms: 2.25 + t = -2.657536453 Solving 2.25 + t = -2.657536453 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = -2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = -2.657536453 + -2.25 t = -2.657536453 + -2.25 Combine like terms: -2.657536453 + -2.25 = -4.907536453 t = -4.907536453 Simplifying t = -4.907536453Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.407536453, -4.907536453}Solution
t = {0.407536453, -4.907536453}
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