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Simplifying 16t2 + 112t + 128 = 0 Reorder the terms: 128 + 112t + 16t2 = 0 Solving 128 + 112t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(8 + 7t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(8 + 7t + t2)' equal to zero and attempt to solve: Simplifying 8 + 7t + t2 = 0 Solving 8 + 7t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + 7t + -8 + t2 = 0 + -8 Reorder the terms: 8 + -8 + 7t + t2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 7t + t2 = 0 + -8 7t + t2 = 0 + -8 Combine like terms: 0 + -8 = -8 7t + t2 = -8 The t term is 7t. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7t + 12.25 + t2 = -8 + 12.25 Reorder the terms: 12.25 + 7t + t2 = -8 + 12.25 Combine like terms: -8 + 12.25 = 4.25 12.25 + 7t + t2 = 4.25 Factor a perfect square on the left side: (t + 3.5)(t + 3.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (t + 3.5) equal to 2.061552813 and -2.061552813.Subproblem 1
t + 3.5 = 2.061552813 Simplifying t + 3.5 = 2.061552813 Reorder the terms: 3.5 + t = 2.061552813 Solving 3.5 + t = 2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = 2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = 2.061552813 + -3.5 t = 2.061552813 + -3.5 Combine like terms: 2.061552813 + -3.5 = -1.438447187 t = -1.438447187 Simplifying t = -1.438447187Subproblem 2
t + 3.5 = -2.061552813 Simplifying t + 3.5 = -2.061552813 Reorder the terms: 3.5 + t = -2.061552813 Solving 3.5 + t = -2.061552813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + t = -2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + t = -2.061552813 + -3.5 t = -2.061552813 + -3.5 Combine like terms: -2.061552813 + -3.5 = -5.561552813 t = -5.561552813 Simplifying t = -5.561552813Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.438447187, -5.561552813}Solution
t = {-1.438447187, -5.561552813}
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