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Simplifying t3 + -30t2 + 236t = 50t + -1t2 Reorder the terms: 236t + -30t2 + t3 = 50t + -1t2 Solving 236t + -30t2 + t3 = 50t + -1t2 Solving for variable 't'. Reorder the terms: 236t + -50t + -30t2 + t2 + t3 = 50t + -1t2 + -50t + t2 Combine like terms: 236t + -50t = 186t 186t + -30t2 + t2 + t3 = 50t + -1t2 + -50t + t2 Combine like terms: -30t2 + t2 = -29t2 186t + -29t2 + t3 = 50t + -1t2 + -50t + t2 Reorder the terms: 186t + -29t2 + t3 = 50t + -50t + -1t2 + t2 Combine like terms: 50t + -50t = 0 186t + -29t2 + t3 = 0 + -1t2 + t2 186t + -29t2 + t3 = -1t2 + t2 Combine like terms: -1t2 + t2 = 0 186t + -29t2 + t3 = 0 Factor out the Greatest Common Factor (GCF), 't'. t(186 + -29t + t2) = 0Subproblem 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 '(186 + -29t + t2)' equal to zero and attempt to solve: Simplifying 186 + -29t + t2 = 0 Solving 186 + -29t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-186' to each side of the equation. 186 + -29t + -186 + t2 = 0 + -186 Reorder the terms: 186 + -186 + -29t + t2 = 0 + -186 Combine like terms: 186 + -186 = 0 0 + -29t + t2 = 0 + -186 -29t + t2 = 0 + -186 Combine like terms: 0 + -186 = -186 -29t + t2 = -186 The t term is -29t. Take half its coefficient (-14.5). Square it (210.25) and add it to both sides. Add '210.25' to each side of the equation. -29t + 210.25 + t2 = -186 + 210.25 Reorder the terms: 210.25 + -29t + t2 = -186 + 210.25 Combine like terms: -186 + 210.25 = 24.25 210.25 + -29t + t2 = 24.25 Factor a perfect square on the left side: (t + -14.5)(t + -14.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (t + -14.5) equal to 4.924428901 and -4.924428901.Subproblem 1
t + -14.5 = 4.924428901 Simplifying t + -14.5 = 4.924428901 Reorder the terms: -14.5 + t = 4.924428901 Solving -14.5 + t = 4.924428901 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + t = 4.924428901 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + t = 4.924428901 + 14.5 t = 4.924428901 + 14.5 Combine like terms: 4.924428901 + 14.5 = 19.424428901 t = 19.424428901 Simplifying t = 19.424428901Subproblem 2
t + -14.5 = -4.924428901 Simplifying t + -14.5 = -4.924428901 Reorder the terms: -14.5 + t = -4.924428901 Solving -14.5 + t = -4.924428901 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + t = -4.924428901 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + t = -4.924428901 + 14.5 t = -4.924428901 + 14.5 Combine like terms: -4.924428901 + 14.5 = 9.575571099 t = 9.575571099 Simplifying t = 9.575571099Solution
The solution to the problem is based on the solutions from the subproblems. t = {19.424428901, 9.575571099}Solution
t = {0, 19.424428901, 9.575571099}
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