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Simplifying 0 = -16t2 + -8t + 64 Reorder the terms: 0 = 64 + -8t + -16t2 Solving 0 = 64 + -8t + -16t2 Solving for variable 't'. Combine like terms: 0 + -64 = -64 -64 + 8t + 16t2 = 64 + -8t + -16t2 + -64 + 8t + 16t2 Reorder the terms: -64 + 8t + 16t2 = 64 + -64 + -8t + 8t + -16t2 + 16t2 Combine like terms: 64 + -64 = 0 -64 + 8t + 16t2 = 0 + -8t + 8t + -16t2 + 16t2 -64 + 8t + 16t2 = -8t + 8t + -16t2 + 16t2 Combine like terms: -8t + 8t = 0 -64 + 8t + 16t2 = 0 + -16t2 + 16t2 -64 + 8t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -64 + 8t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-8 + t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-8 + t + 2t2)' equal to zero and attempt to solve: Simplifying -8 + t + 2t2 = 0 Solving -8 + t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4 + 0.5t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.5t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + 0.5t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.5t + t2 = 0 + 4 0.5t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.5t + t2 = 4 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5t + 0.25 + t2 = 4 + 0.25 Reorder the terms: 0.25 + 0.5t + t2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + 0.5t + t2 = 4.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (t + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
t + 0.5 = 2.061552813 Simplifying t + 0.5 = 2.061552813 Reorder the terms: 0.5 + t = 2.061552813 Solving 0.5 + t = 2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 2.061552813 + -0.5 t = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 t = 1.561552813 Simplifying t = 1.561552813Subproblem 2
t + 0.5 = -2.061552813 Simplifying t + 0.5 = -2.061552813 Reorder the terms: 0.5 + t = -2.061552813 Solving 0.5 + t = -2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -2.061552813 + -0.5 t = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 t = -2.561552813 Simplifying t = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.561552813, -2.561552813}Solution
t = {1.561552813, -2.561552813}
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