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Simplifying 0 = -16t2 + 32t + 12 Reorder the terms: 0 = 12 + 32t + -16t2 Solving 0 = 12 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -32t + 16t2 = 12 + 32t + -16t2 + -12 + -32t + 16t2 Reorder the terms: -12 + -32t + 16t2 = 12 + -12 + 32t + -32t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -12 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -12 + -32t + 16t2 = 0 + -16t2 + 16t2 -12 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -8t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -8t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + -8t + 4t2 = 0 Solving -3 + -8t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -2t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -2t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -2t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -2t + t2 = 0 + 0.75 -2t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -2t + t2 = 0.75 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 0.75 + 1 Reorder the terms: 1 + -2t + t2 = 0.75 + 1 Combine like terms: 0.75 + 1 = 1.75 1 + -2t + t2 = 1.75 Factor a perfect square on the left side: (t + -1)(t + -1) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (t + -1) equal to 1.322875656 and -1.322875656.Subproblem 1
t + -1 = 1.322875656 Simplifying t + -1 = 1.322875656 Reorder the terms: -1 + t = 1.322875656 Solving -1 + t = 1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 1.322875656 + 1 Combine like terms: -1 + 1 = 0 0 + t = 1.322875656 + 1 t = 1.322875656 + 1 Combine like terms: 1.322875656 + 1 = 2.322875656 t = 2.322875656 Simplifying t = 2.322875656Subproblem 2
t + -1 = -1.322875656 Simplifying t + -1 = -1.322875656 Reorder the terms: -1 + t = -1.322875656 Solving -1 + t = -1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -1.322875656 + 1 Combine like terms: -1 + 1 = 0 0 + t = -1.322875656 + 1 t = -1.322875656 + 1 Combine like terms: -1.322875656 + 1 = -0.322875656 t = -0.322875656 Simplifying t = -0.322875656Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.322875656, -0.322875656}Solution
t = {2.322875656, -0.322875656}
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