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Simplifying 3t4 + 4t3 + -6t2 = 0 Reorder the terms: -6t2 + 4t3 + 3t4 = 0 Solving -6t2 + 4t3 + 3t4 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), 't2'. t2(-6 + 4t + 3t2) = 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 '(-6 + 4t + 3t2)' equal to zero and attempt to solve: Simplifying -6 + 4t + 3t2 = 0 Solving -6 + 4t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 1.333333333t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.333333333t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 1.333333333t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.333333333t + t2 = 0 + 2 1.333333333t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.333333333t + t2 = 2 The t term is 1.333333333t. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333t + 0.4444444442 + t2 = 2 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333t + t2 = 2 + 0.4444444442 Combine like terms: 2 + 0.4444444442 = 2.4444444442 0.4444444442 + 1.333333333t + t2 = 2.4444444442 Factor a perfect square on the left side: (t + 0.6666666665)(t + 0.6666666665) = 2.4444444442 Calculate the square root of the right side: 1.56347192 Break this problem into two subproblems by setting (t + 0.6666666665) equal to 1.56347192 and -1.56347192.Subproblem 1
t + 0.6666666665 = 1.56347192 Simplifying t + 0.6666666665 = 1.56347192 Reorder the terms: 0.6666666665 + t = 1.56347192 Solving 0.6666666665 + t = 1.56347192 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + t = 1.56347192 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + t = 1.56347192 + -0.6666666665 t = 1.56347192 + -0.6666666665 Combine like terms: 1.56347192 + -0.6666666665 = 0.8968052535 t = 0.8968052535 Simplifying t = 0.8968052535Subproblem 2
t + 0.6666666665 = -1.56347192 Simplifying t + 0.6666666665 = -1.56347192 Reorder the terms: 0.6666666665 + t = -1.56347192 Solving 0.6666666665 + t = -1.56347192 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + t = -1.56347192 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + t = -1.56347192 + -0.6666666665 t = -1.56347192 + -0.6666666665 Combine like terms: -1.56347192 + -0.6666666665 = -2.2301385865 t = -2.2301385865 Simplifying t = -2.2301385865Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.8968052535, -2.2301385865}Solution
t = {0, 0.8968052535, -2.2301385865}
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