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Simplifying 12t2 + 48t + -20 = 0 Reorder the terms: -20 + 48t + 12t2 = 0 Solving -20 + 48t + 12t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(-5 + 12t + 3t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-5 + 12t + 3t2)' equal to zero and attempt to solve: Simplifying -5 + 12t + 3t2 = 0 Solving -5 + 12t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + 4t + t2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 4t + 1.666666667 + t2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 4t + t2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 4t + t2 = 0 + 1.666666667 4t + t2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 4t + t2 = 1.666666667 The t term is 4t. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4t + 4 + t2 = 1.666666667 + 4 Reorder the terms: 4 + 4t + t2 = 1.666666667 + 4 Combine like terms: 1.666666667 + 4 = 5.666666667 4 + 4t + t2 = 5.666666667 Factor a perfect square on the left side: (t + 2)(t + 2) = 5.666666667 Calculate the square root of the right side: 2.380476143 Break this problem into two subproblems by setting (t + 2) equal to 2.380476143 and -2.380476143.Subproblem 1
t + 2 = 2.380476143 Simplifying t + 2 = 2.380476143 Reorder the terms: 2 + t = 2.380476143 Solving 2 + t = 2.380476143 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + t = 2.380476143 + -2 Combine like terms: 2 + -2 = 0 0 + t = 2.380476143 + -2 t = 2.380476143 + -2 Combine like terms: 2.380476143 + -2 = 0.380476143 t = 0.380476143 Simplifying t = 0.380476143Subproblem 2
t + 2 = -2.380476143 Simplifying t + 2 = -2.380476143 Reorder the terms: 2 + t = -2.380476143 Solving 2 + t = -2.380476143 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + t = -2.380476143 + -2 Combine like terms: 2 + -2 = 0 0 + t = -2.380476143 + -2 t = -2.380476143 + -2 Combine like terms: -2.380476143 + -2 = -4.380476143 t = -4.380476143 Simplifying t = -4.380476143Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.380476143, -4.380476143}Solution
t = {0.380476143, -4.380476143}
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