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Simplifying 14t4 + 9t3 + -42t2 = 0 Reorder the terms: -42t2 + 9t3 + 14t4 = 0 Solving -42t2 + 9t3 + 14t4 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't2'. t2(-42 + 9t + 14t2) = 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 '(-42 + 9t + 14t2)' equal to zero and attempt to solve: Simplifying -42 + 9t + 14t2 = 0 Solving -42 + 9t + 14t2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -3 + 0.6428571429t + t2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.6428571429t + 3 + t2 = 0 + 3 Reorder the terms: -3 + 3 + 0.6428571429t + t2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.6428571429t + t2 = 0 + 3 0.6428571429t + t2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.6428571429t + t2 = 3 The t term is 0.6428571429t. Take half its coefficient (0.3214285715). Square it (0.1033163266) and add it to both sides. Add '0.1033163266' to each side of the equation. 0.6428571429t + 0.1033163266 + t2 = 3 + 0.1033163266 Reorder the terms: 0.1033163266 + 0.6428571429t + t2 = 3 + 0.1033163266 Combine like terms: 3 + 0.1033163266 = 3.1033163266 0.1033163266 + 0.6428571429t + t2 = 3.1033163266 Factor a perfect square on the left side: (t + 0.3214285715)(t + 0.3214285715) = 3.1033163266 Calculate the square root of the right side: 1.761623208 Break this problem into two subproblems by setting (t + 0.3214285715) equal to 1.761623208 and -1.761623208.Subproblem 1
t + 0.3214285715 = 1.761623208 Simplifying t + 0.3214285715 = 1.761623208 Reorder the terms: 0.3214285715 + t = 1.761623208 Solving 0.3214285715 + t = 1.761623208 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3214285715' to each side of the equation. 0.3214285715 + -0.3214285715 + t = 1.761623208 + -0.3214285715 Combine like terms: 0.3214285715 + -0.3214285715 = 0.0000000000 0.0000000000 + t = 1.761623208 + -0.3214285715 t = 1.761623208 + -0.3214285715 Combine like terms: 1.761623208 + -0.3214285715 = 1.4401946365 t = 1.4401946365 Simplifying t = 1.4401946365Subproblem 2
t + 0.3214285715 = -1.761623208 Simplifying t + 0.3214285715 = -1.761623208 Reorder the terms: 0.3214285715 + t = -1.761623208 Solving 0.3214285715 + t = -1.761623208 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3214285715' to each side of the equation. 0.3214285715 + -0.3214285715 + t = -1.761623208 + -0.3214285715 Combine like terms: 0.3214285715 + -0.3214285715 = 0.0000000000 0.0000000000 + t = -1.761623208 + -0.3214285715 t = -1.761623208 + -0.3214285715 Combine like terms: -1.761623208 + -0.3214285715 = -2.0830517795 t = -2.0830517795 Simplifying t = -2.0830517795Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.4401946365, -2.0830517795}Solution
t = {0, 1.4401946365, -2.0830517795}
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