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Simplifying 30 + -16t2 + 56t = 6 Reorder the terms: 30 + 56t + -16t2 = 6 Solving 30 + 56t + -16t2 = 6 Solving for variable 't'. Reorder the terms: 30 + -6 + 56t + -16t2 = 6 + -6 Combine like terms: 30 + -6 = 24 24 + 56t + -16t2 = 6 + -6 Combine like terms: 6 + -6 = 0 24 + 56t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(3 + 7t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(3 + 7t + -2t2)' equal to zero and attempt to solve: Simplifying 3 + 7t + -2t2 = 0 Solving 3 + 7t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -3.5t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -3.5t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -3.5t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -3.5t + t2 = 0 + 1.5 -3.5t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -3.5t + t2 = 1.5 The t term is -3.5t. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5t + 3.0625 + t2 = 1.5 + 3.0625 Reorder the terms: 3.0625 + -3.5t + t2 = 1.5 + 3.0625 Combine like terms: 1.5 + 3.0625 = 4.5625 3.0625 + -3.5t + t2 = 4.5625 Factor a perfect square on the left side: (t + -1.75)(t + -1.75) = 4.5625 Calculate the square root of the right side: 2.136000936 Break this problem into two subproblems by setting (t + -1.75) equal to 2.136000936 and -2.136000936.Subproblem 1
t + -1.75 = 2.136000936 Simplifying t + -1.75 = 2.136000936 Reorder the terms: -1.75 + t = 2.136000936 Solving -1.75 + t = 2.136000936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = 2.136000936 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = 2.136000936 + 1.75 t = 2.136000936 + 1.75 Combine like terms: 2.136000936 + 1.75 = 3.886000936 t = 3.886000936 Simplifying t = 3.886000936Subproblem 2
t + -1.75 = -2.136000936 Simplifying t + -1.75 = -2.136000936 Reorder the terms: -1.75 + t = -2.136000936 Solving -1.75 + t = -2.136000936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = -2.136000936 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = -2.136000936 + 1.75 t = -2.136000936 + 1.75 Combine like terms: -2.136000936 + 1.75 = -0.386000936 t = -0.386000936 Simplifying t = -0.386000936Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.886000936, -0.386000936}Solution
t = {3.886000936, -0.386000936}
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