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Simplifying 0 = -16t2 + 24t + 12 Reorder the terms: 0 = 12 + 24t + -16t2 Solving 0 = 12 + 24t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -24t + 16t2 = 12 + 24t + -16t2 + -12 + -24t + 16t2 Reorder the terms: -12 + -24t + 16t2 = 12 + -12 + 24t + -24t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -24t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -12 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -12 + -24t + 16t2 = 0 + -16t2 + 16t2 -12 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -24t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -6t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -6t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + -6t + 4t2 = 0 Solving -3 + -6t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.5t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.5t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.5t + t2 = 0 + 0.75 -1.5t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.5t + t2 = 0.75 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 = 0.75 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + -1.5t + t2 = 1.3125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (t + -0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
t + -0.75 = 1.145643924 Simplifying t + -0.75 = 1.145643924 Reorder the terms: -0.75 + t = 1.145643924 Solving -0.75 + t = 1.145643924 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 = 1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.145643924 + 0.75 t = 1.145643924 + 0.75 Combine like terms: 1.145643924 + 0.75 = 1.895643924 t = 1.895643924 Simplifying t = 1.895643924Subproblem 2
t + -0.75 = -1.145643924 Simplifying t + -0.75 = -1.145643924 Reorder the terms: -0.75 + t = -1.145643924 Solving -0.75 + t = -1.145643924 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 = -1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.145643924 + 0.75 t = -1.145643924 + 0.75 Combine like terms: -1.145643924 + 0.75 = -0.395643924 t = -0.395643924 Simplifying t = -0.395643924Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.895643924, -0.395643924}Solution
t = {1.895643924, -0.395643924}
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