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Simplifying 14t2 + 6t3 + -4t3 + -2t2 + -3t2 + -5t4 = 0 Reorder the terms: 14t2 + -2t2 + -3t2 + 6t3 + -4t3 + -5t4 = 0 Combine like terms: 14t2 + -2t2 = 12t2 12t2 + -3t2 + 6t3 + -4t3 + -5t4 = 0 Combine like terms: 12t2 + -3t2 = 9t2 9t2 + 6t3 + -4t3 + -5t4 = 0 Combine like terms: 6t3 + -4t3 = 2t3 9t2 + 2t3 + -5t4 = 0 Solving 9t2 + 2t3 + -5t4 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't2'. t2(9 + 2t + -5t2) = 0Subproblem 1
Set the factor 't2' equal to zero and attempt to solve: Simplifying t2 = 0 Solving t2 = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t2 = 0 Take the square root of each side: t = {0}Subproblem 2
Set the factor '(9 + 2t + -5t2)' equal to zero and attempt to solve: Simplifying 9 + 2t + -5t2 = 0 Solving 9 + 2t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1.8 + -0.4t + t2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + -0.4t + 1.8 + t2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + -0.4t + t2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + -0.4t + t2 = 0 + 1.8 -0.4t + t2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 -0.4t + t2 = 1.8 The t term is -0.4t. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4t + 0.04 + t2 = 1.8 + 0.04 Reorder the terms: 0.04 + -0.4t + t2 = 1.8 + 0.04 Combine like terms: 1.8 + 0.04 = 1.84 0.04 + -0.4t + t2 = 1.84 Factor a perfect square on the left side: (t + -0.2)(t + -0.2) = 1.84 Calculate the square root of the right side: 1.356465997 Break this problem into two subproblems by setting (t + -0.2) equal to 1.356465997 and -1.356465997.Subproblem 1
t + -0.2 = 1.356465997 Simplifying t + -0.2 = 1.356465997 Reorder the terms: -0.2 + t = 1.356465997 Solving -0.2 + t = 1.356465997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + t = 1.356465997 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + t = 1.356465997 + 0.2 t = 1.356465997 + 0.2 Combine like terms: 1.356465997 + 0.2 = 1.556465997 t = 1.556465997 Simplifying t = 1.556465997Subproblem 2
t + -0.2 = -1.356465997 Simplifying t + -0.2 = -1.356465997 Reorder the terms: -0.2 + t = -1.356465997 Solving -0.2 + t = -1.356465997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + t = -1.356465997 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + t = -1.356465997 + 0.2 t = -1.356465997 + 0.2 Combine like terms: -1.356465997 + 0.2 = -1.156465997 t = -1.156465997 Simplifying t = -1.156465997Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.556465997, -1.156465997}Solution
t = {0, 1.556465997, -1.156465997}
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