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Simplifying 36t2 + 162t + 16 = 0 Reorder the terms: 16 + 162t + 36t2 = 0 Solving 16 + 162t + 36t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(8 + 81t + 18t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + 81t + 18t2)' equal to zero and attempt to solve: Simplifying 8 + 81t + 18t2 = 0 Solving 8 + 81t + 18t2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.4444444444 + 4.5t + t2 = 0 Move the constant term to the right: Add '-0.4444444444' to each side of the equation. 0.4444444444 + 4.5t + -0.4444444444 + t2 = 0 + -0.4444444444 Reorder the terms: 0.4444444444 + -0.4444444444 + 4.5t + t2 = 0 + -0.4444444444 Combine like terms: 0.4444444444 + -0.4444444444 = 0.0000000000 0.0000000000 + 4.5t + t2 = 0 + -0.4444444444 4.5t + t2 = 0 + -0.4444444444 Combine like terms: 0 + -0.4444444444 = -0.4444444444 4.5t + t2 = -0.4444444444 The t term is 4.5t. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5t + 5.0625 + t2 = -0.4444444444 + 5.0625 Reorder the terms: 5.0625 + 4.5t + t2 = -0.4444444444 + 5.0625 Combine like terms: -0.4444444444 + 5.0625 = 4.6180555556 5.0625 + 4.5t + t2 = 4.6180555556 Factor a perfect square on the left side: (t + 2.25)(t + 2.25) = 4.6180555556 Calculate the square root of the right side: 2.14896616 Break this problem into two subproblems by setting (t + 2.25) equal to 2.14896616 and -2.14896616.Subproblem 1
t + 2.25 = 2.14896616 Simplifying t + 2.25 = 2.14896616 Reorder the terms: 2.25 + t = 2.14896616 Solving 2.25 + t = 2.14896616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = 2.14896616 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = 2.14896616 + -2.25 t = 2.14896616 + -2.25 Combine like terms: 2.14896616 + -2.25 = -0.10103384 t = -0.10103384 Simplifying t = -0.10103384Subproblem 2
t + 2.25 = -2.14896616 Simplifying t + 2.25 = -2.14896616 Reorder the terms: 2.25 + t = -2.14896616 Solving 2.25 + t = -2.14896616 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + t = -2.14896616 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + t = -2.14896616 + -2.25 t = -2.14896616 + -2.25 Combine like terms: -2.14896616 + -2.25 = -4.39896616 t = -4.39896616 Simplifying t = -4.39896616Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.10103384, -4.39896616}Solution
t = {-0.10103384, -4.39896616}
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