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Simplifying 4t2 + 24t + 14 = 0 Reorder the terms: 14 + 24t + 4t2 = 0 Solving 14 + 24t + 4t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + 12t + 2t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + 12t + 2t2)' equal to zero and attempt to solve: Simplifying 7 + 12t + 2t2 = 0 Solving 7 + 12t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + 6t + t2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 6t + -3.5 + t2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 6t + t2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 6t + t2 = 0 + -3.5 6t + t2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 6t + t2 = -3.5 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 = -3.5 + 9 Reorder the terms: 9 + 6t + t2 = -3.5 + 9 Combine like terms: -3.5 + 9 = 5.5 9 + 6t + t2 = 5.5 Factor a perfect square on the left side: (t + 3)(t + 3) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (t + 3) equal to 2.34520788 and -2.34520788.Subproblem 1
t + 3 = 2.34520788 Simplifying t + 3 = 2.34520788 Reorder the terms: 3 + t = 2.34520788 Solving 3 + t = 2.34520788 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 = 2.34520788 + -3 Combine like terms: 3 + -3 = 0 0 + t = 2.34520788 + -3 t = 2.34520788 + -3 Combine like terms: 2.34520788 + -3 = -0.65479212 t = -0.65479212 Simplifying t = -0.65479212Subproblem 2
t + 3 = -2.34520788 Simplifying t + 3 = -2.34520788 Reorder the terms: 3 + t = -2.34520788 Solving 3 + t = -2.34520788 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 = -2.34520788 + -3 Combine like terms: 3 + -3 = 0 0 + t = -2.34520788 + -3 t = -2.34520788 + -3 Combine like terms: -2.34520788 + -3 = -5.34520788 t = -5.34520788 Simplifying t = -5.34520788Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.65479212, -5.34520788}Solution
t = {-0.65479212, -5.34520788}
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