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Simplifying -12t2 + 36t + 42 = 0 Reorder the terms: 42 + 36t + -12t2 = 0 Solving 42 + 36t + -12t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '6'. 6(7 + 6t + -2t2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(7 + 6t + -2t2)' equal to zero and attempt to solve: Simplifying 7 + 6t + -2t2 = 0 Solving 7 + 6t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3.5 + -3t + t2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -3t + 3.5 + t2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -3t + t2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -3t + t2 = 0 + 3.5 -3t + t2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -3t + t2 = 3.5 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 3.5 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 3.5 + 2.25 Combine like terms: 3.5 + 2.25 = 5.75 2.25 + -3t + t2 = 5.75 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (t + -1.5) equal to 2.397915762 and -2.397915762.Subproblem 1
t + -1.5 = 2.397915762 Simplifying t + -1.5 = 2.397915762 Reorder the terms: -1.5 + t = 2.397915762 Solving -1.5 + t = 2.397915762 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2.397915762 + 1.5 t = 2.397915762 + 1.5 Combine like terms: 2.397915762 + 1.5 = 3.897915762 t = 3.897915762 Simplifying t = 3.897915762Subproblem 2
t + -1.5 = -2.397915762 Simplifying t + -1.5 = -2.397915762 Reorder the terms: -1.5 + t = -2.397915762 Solving -1.5 + t = -2.397915762 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2.397915762 + 1.5 t = -2.397915762 + 1.5 Combine like terms: -2.397915762 + 1.5 = -0.897915762 t = -0.897915762 Simplifying t = -0.897915762Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.897915762, -0.897915762}Solution
t = {3.897915762, -0.897915762}
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