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Simplifying 12t3 + 60t2 + -36t = 0 Reorder the terms: -36t + 60t2 + 12t3 = 0 Solving -36t + 60t2 + 12t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '12t'. 12t(-3 + 5t + t2) = 0 Ignore the factor 12.Subproblem 1
Set the factor 't' equal to zero and attempt to solve: Simplifying t = 0 Solving t = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t = 0Subproblem 2
Set the factor '(-3 + 5t + t2)' equal to zero and attempt to solve: Simplifying -3 + 5t + t2 = 0 Solving -3 + 5t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 5t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + 5t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 5t + t2 = 0 + 3 5t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 5t + t2 = 3 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = 3 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 3 + 6.25 Combine like terms: 3 + 6.25 = 9.25 6.25 + 5t + t2 = 9.25 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (t + 2.5) equal to 3.041381265 and -3.041381265.Subproblem 1
t + 2.5 = 3.041381265 Simplifying t + 2.5 = 3.041381265 Reorder the terms: 2.5 + t = 3.041381265 Solving 2.5 + t = 3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 3.041381265 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 3.041381265 + -2.5 t = 3.041381265 + -2.5 Combine like terms: 3.041381265 + -2.5 = 0.541381265 t = 0.541381265 Simplifying t = 0.541381265Subproblem 2
t + 2.5 = -3.041381265 Simplifying t + 2.5 = -3.041381265 Reorder the terms: 2.5 + t = -3.041381265 Solving 2.5 + t = -3.041381265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -3.041381265 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -3.041381265 + -2.5 t = -3.041381265 + -2.5 Combine like terms: -3.041381265 + -2.5 = -5.541381265 t = -5.541381265 Simplifying t = -5.541381265Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.541381265, -5.541381265}Solution
t = {0, 0.541381265, -5.541381265}
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