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Simplifying 16t2 + 96t + 12 = 0 Reorder the terms: 12 + 96t + 16t2 = 0 Solving 12 + 96t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(3 + 24t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(3 + 24t + 4t2)' equal to zero and attempt to solve: Simplifying 3 + 24t + 4t2 = 0 Solving 3 + 24t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 6t + t2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 6t + -0.75 + t2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 6t + t2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 6t + t2 = 0 + -0.75 6t + t2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 6t + t2 = -0.75 The t term is 6t. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6t + 9 + t2 = -0.75 + 9 Reorder the terms: 9 + 6t + t2 = -0.75 + 9 Combine like terms: -0.75 + 9 = 8.25 9 + 6t + t2 = 8.25 Factor a perfect square on the left side: (t + 3)(t + 3) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (t + 3) equal to 2.872281323 and -2.872281323.Subproblem 1
t + 3 = 2.872281323 Simplifying t + 3 = 2.872281323 Reorder the terms: 3 + t = 2.872281323 Solving 3 + t = 2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = 2.872281323 + -3 Combine like terms: 3 + -3 = 0 0 + t = 2.872281323 + -3 t = 2.872281323 + -3 Combine like terms: 2.872281323 + -3 = -0.127718677 t = -0.127718677 Simplifying t = -0.127718677Subproblem 2
t + 3 = -2.872281323 Simplifying t + 3 = -2.872281323 Reorder the terms: 3 + t = -2.872281323 Solving 3 + t = -2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = -2.872281323 + -3 Combine like terms: 3 + -3 = 0 0 + t = -2.872281323 + -3 t = -2.872281323 + -3 Combine like terms: -2.872281323 + -3 = -5.872281323 t = -5.872281323 Simplifying t = -5.872281323Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.127718677, -5.872281323}Solution
t = {-0.127718677, -5.872281323}
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