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Simplifying 0 = -16t2 + 24t + 144 Reorder the terms: 0 = 144 + 24t + -16t2 Solving 0 = 144 + 24t + -16t2 Solving for variable 't'. Combine like terms: 0 + -144 = -144 -144 + -24t + 16t2 = 144 + 24t + -16t2 + -144 + -24t + 16t2 Reorder the terms: -144 + -24t + 16t2 = 144 + -144 + 24t + -24t + -16t2 + 16t2 Combine like terms: 144 + -144 = 0 -144 + -24t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -144 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -144 + -24t + 16t2 = 0 + -16t2 + 16t2 -144 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -144 + -24t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-18 + -3t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-18 + -3t + 2t2)' equal to zero and attempt to solve: Simplifying -18 + -3t + 2t2 = 0 Solving -18 + -3t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + -1.5t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -1.5t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + -1.5t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -1.5t + t2 = 0 + 9 -1.5t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 -1.5t + t2 = 9 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 = 9 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 9 + 0.5625 Combine like terms: 9 + 0.5625 = 9.5625 0.5625 + -1.5t + t2 = 9.5625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (t + -0.75) equal to 3.092329219 and -3.092329219.Subproblem 1
t + -0.75 = 3.092329219 Simplifying t + -0.75 = 3.092329219 Reorder the terms: -0.75 + t = 3.092329219 Solving -0.75 + t = 3.092329219 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 = 3.092329219 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 3.092329219 + 0.75 t = 3.092329219 + 0.75 Combine like terms: 3.092329219 + 0.75 = 3.842329219 t = 3.842329219 Simplifying t = 3.842329219Subproblem 2
t + -0.75 = -3.092329219 Simplifying t + -0.75 = -3.092329219 Reorder the terms: -0.75 + t = -3.092329219 Solving -0.75 + t = -3.092329219 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 = -3.092329219 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -3.092329219 + 0.75 t = -3.092329219 + 0.75 Combine like terms: -3.092329219 + 0.75 = -2.342329219 t = -2.342329219 Simplifying t = -2.342329219Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.842329219, -2.342329219}Solution
t = {3.842329219, -2.342329219}
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