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Simplifying 0 = -16t2 + 24t + 30 Reorder the terms: 0 = 30 + 24t + -16t2 Solving 0 = 30 + 24t + -16t2 Solving for variable 't'. Combine like terms: 0 + -30 = -30 -30 + -24t + 16t2 = 30 + 24t + -16t2 + -30 + -24t + 16t2 Reorder the terms: -30 + -24t + 16t2 = 30 + -30 + 24t + -24t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 -30 + -24t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -30 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -30 + -24t + 16t2 = 0 + -16t2 + 16t2 -30 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -30 + -24t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + -12t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + -12t + 8t2)' equal to zero and attempt to solve: Simplifying -15 + -12t + 8t2 = 0 Solving -15 + -12t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.875 + -1.5t + t2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + -1.5t + 1.875 + t2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + -1.5t + t2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + -1.5t + t2 = 0 + 1.875 -1.5t + t2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 -1.5t + t2 = 1.875 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 = 1.875 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 1.875 + 0.5625 Combine like terms: 1.875 + 0.5625 = 2.4375 0.5625 + -1.5t + t2 = 2.4375 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 2.4375 Calculate the square root of the right side: 1.5612495 Break this problem into two subproblems by setting (t + -0.75) equal to 1.5612495 and -1.5612495.Subproblem 1
t + -0.75 = 1.5612495 Simplifying t + -0.75 = 1.5612495 Reorder the terms: -0.75 + t = 1.5612495 Solving -0.75 + t = 1.5612495 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 = 1.5612495 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.5612495 + 0.75 t = 1.5612495 + 0.75 Combine like terms: 1.5612495 + 0.75 = 2.3112495 t = 2.3112495 Simplifying t = 2.3112495Subproblem 2
t + -0.75 = -1.5612495 Simplifying t + -0.75 = -1.5612495 Reorder the terms: -0.75 + t = -1.5612495 Solving -0.75 + t = -1.5612495 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 = -1.5612495 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.5612495 + 0.75 t = -1.5612495 + 0.75 Combine like terms: -1.5612495 + 0.75 = -0.8112495 t = -0.8112495 Simplifying t = -0.8112495Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.3112495, -0.8112495}Solution
t = {2.3112495, -0.8112495}
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