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Simplifying -32t2 + 48t + 16 = 0 Reorder the terms: 16 + 48t + -32t2 = 0 Solving 16 + 48t + -32t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(1 + 3t + -2t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(1 + 3t + -2t2)' equal to zero and attempt to solve: Simplifying 1 + 3t + -2t2 = 0 Solving 1 + 3t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.5t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.5t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.5t + t2 = 0 + 0.5 -1.5t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.5t + t2 = 0.5 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.5 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + -1.5t + t2 = 1.0625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (t + -0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
t + -0.75 = 1.030776406 Simplifying t + -0.75 = 1.030776406 Reorder the terms: -0.75 + t = 1.030776406 Solving -0.75 + t = 1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.030776406 + 0.75 t = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 t = 1.780776406 Simplifying t = 1.780776406Subproblem 2
t + -0.75 = -1.030776406 Simplifying t + -0.75 = -1.030776406 Reorder the terms: -0.75 + t = -1.030776406 Solving -0.75 + t = -1.030776406 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.030776406 + 0.75 t = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 t = -0.280776406 Simplifying t = -0.280776406Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.780776406, -0.280776406}Solution
t = {1.780776406, -0.280776406}
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