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Simplifying 4t2 + 4t + -14t3 = 0 Reorder the terms: 4t + 4t2 + -14t3 = 0 Solving 4t + 4t2 + -14t3 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2t'. 2t(2 + 2t + -7t2) = 0 Ignore the factor 2.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 '(2 + 2t + -7t2)' equal to zero and attempt to solve: Simplifying 2 + 2t + -7t2 = 0 Solving 2 + 2t + -7t2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -0.2857142857 + -0.2857142857t + t2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -0.2857142857t + 0.2857142857 + t2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -0.2857142857t + t2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -0.2857142857t + t2 = 0 + 0.2857142857 -0.2857142857t + t2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -0.2857142857t + t2 = 0.2857142857 The t term is -0.2857142857t. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857t + 0.02040816328 + t2 = 0.2857142857 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857t + t2 = 0.2857142857 + 0.02040816328 Combine like terms: 0.2857142857 + 0.02040816328 = 0.30612244898 0.02040816328 + -0.2857142857t + t2 = 0.30612244898 Factor a perfect square on the left side: (t + -0.1428571429)(t + -0.1428571429) = 0.30612244898 Calculate the square root of the right side: 0.553283335 Break this problem into two subproblems by setting (t + -0.1428571429) equal to 0.553283335 and -0.553283335.Subproblem 1
t + -0.1428571429 = 0.553283335 Simplifying t + -0.1428571429 = 0.553283335 Reorder the terms: -0.1428571429 + t = 0.553283335 Solving -0.1428571429 + t = 0.553283335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + t = 0.553283335 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + t = 0.553283335 + 0.1428571429 t = 0.553283335 + 0.1428571429 Combine like terms: 0.553283335 + 0.1428571429 = 0.6961404779 t = 0.6961404779 Simplifying t = 0.6961404779Subproblem 2
t + -0.1428571429 = -0.553283335 Simplifying t + -0.1428571429 = -0.553283335 Reorder the terms: -0.1428571429 + t = -0.553283335 Solving -0.1428571429 + t = -0.553283335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + t = -0.553283335 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + t = -0.553283335 + 0.1428571429 t = -0.553283335 + 0.1428571429 Combine like terms: -0.553283335 + 0.1428571429 = -0.4104261921 t = -0.4104261921 Simplifying t = -0.4104261921Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.6961404779, -0.4104261921}Solution
t = {0, 0.6961404779, -0.4104261921}
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