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Simplifying 16t2 + -192t + 96 = 0 Reorder the terms: 96 + -192t + 16t2 = 0 Solving 96 + -192t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(6 + -12t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(6 + -12t + t2)' equal to zero and attempt to solve: Simplifying 6 + -12t + t2 = 0 Solving 6 + -12t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-6' to each side of the equation. 6 + -12t + -6 + t2 = 0 + -6 Reorder the terms: 6 + -6 + -12t + t2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -12t + t2 = 0 + -6 -12t + t2 = 0 + -6 Combine like terms: 0 + -6 = -6 -12t + t2 = -6 The t term is -12t. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12t + 36 + t2 = -6 + 36 Reorder the terms: 36 + -12t + t2 = -6 + 36 Combine like terms: -6 + 36 = 30 36 + -12t + t2 = 30 Factor a perfect square on the left side: (t + -6)(t + -6) = 30 Calculate the square root of the right side: 5.477225575 Break this problem into two subproblems by setting (t + -6) equal to 5.477225575 and -5.477225575.Subproblem 1
t + -6 = 5.477225575 Simplifying t + -6 = 5.477225575 Reorder the terms: -6 + t = 5.477225575 Solving -6 + t = 5.477225575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = 5.477225575 + 6 Combine like terms: -6 + 6 = 0 0 + t = 5.477225575 + 6 t = 5.477225575 + 6 Combine like terms: 5.477225575 + 6 = 11.477225575 t = 11.477225575 Simplifying t = 11.477225575Subproblem 2
t + -6 = -5.477225575 Simplifying t + -6 = -5.477225575 Reorder the terms: -6 + t = -5.477225575 Solving -6 + t = -5.477225575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = -5.477225575 + 6 Combine like terms: -6 + 6 = 0 0 + t = -5.477225575 + 6 t = -5.477225575 + 6 Combine like terms: -5.477225575 + 6 = 0.522774425 t = 0.522774425 Simplifying t = 0.522774425Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.477225575, 0.522774425}Solution
t = {11.477225575, 0.522774425}
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