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Simplifying 32t2 + 16t = 480 Reorder the terms: 16t + 32t2 = 480 Solving 16t + 32t2 = 480 Solving for variable 't'. Reorder the terms: -480 + 16t + 32t2 = 480 + -480 Combine like terms: 480 + -480 = 0 -480 + 16t + 32t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-30 + t + 2t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-30 + t + 2t2)' equal to zero and attempt to solve: Simplifying -30 + t + 2t2 = 0 Solving -30 + t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15 + 0.5t + t2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 0.5t + 15 + t2 = 0 + 15 Reorder the terms: -15 + 15 + 0.5t + t2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 0.5t + t2 = 0 + 15 0.5t + t2 = 0 + 15 Combine like terms: 0 + 15 = 15 0.5t + t2 = 15 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5t + 0.25 + t2 = 15 + 0.25 Reorder the terms: 0.25 + 0.5t + t2 = 15 + 0.25 Combine like terms: 15 + 0.25 = 15.25 0.25 + 0.5t + t2 = 15.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (t + 0.5) equal to 3.905124838 and -3.905124838.Subproblem 1
t + 0.5 = 3.905124838 Simplifying t + 0.5 = 3.905124838 Reorder the terms: 0.5 + t = 3.905124838 Solving 0.5 + t = 3.905124838 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 3.905124838 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 3.905124838 + -0.5 t = 3.905124838 + -0.5 Combine like terms: 3.905124838 + -0.5 = 3.405124838 t = 3.405124838 Simplifying t = 3.405124838Subproblem 2
t + 0.5 = -3.905124838 Simplifying t + 0.5 = -3.905124838 Reorder the terms: 0.5 + t = -3.905124838 Solving 0.5 + t = -3.905124838 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -3.905124838 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -3.905124838 + -0.5 t = -3.905124838 + -0.5 Combine like terms: -3.905124838 + -0.5 = -4.405124838 t = -4.405124838 Simplifying t = -4.405124838Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.405124838, -4.405124838}Solution
t = {3.405124838, -4.405124838}
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