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Simplifying 15t4 + -40t3 + -90t2 = 0 Reorder the terms: -90t2 + -40t3 + 15t4 = 0 Solving -90t2 + -40t3 + 15t4 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5t2'. 5t2(-18 + -8t + 3t2) = 0 Ignore the factor 5.Subproblem 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 '(-18 + -8t + 3t2)' equal to zero and attempt to solve: Simplifying -18 + -8t + 3t2 = 0 Solving -18 + -8t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6 + -2.666666667t + t2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -2.666666667t + 6 + t2 = 0 + 6 Reorder the terms: -6 + 6 + -2.666666667t + t2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -2.666666667t + t2 = 0 + 6 -2.666666667t + t2 = 0 + 6 Combine like terms: 0 + 6 = 6 -2.666666667t + t2 = 6 The t term is -2.666666667t. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667t + 1.777777780 + t2 = 6 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667t + t2 = 6 + 1.777777780 Combine like terms: 6 + 1.777777780 = 7.77777778 1.777777780 + -2.666666667t + t2 = 7.77777778 Factor a perfect square on the left side: (t + -1.333333334)(t + -1.333333334) = 7.77777778 Calculate the square root of the right side: 2.788866756 Break this problem into two subproblems by setting (t + -1.333333334) equal to 2.788866756 and -2.788866756.Subproblem 1
t + -1.333333334 = 2.788866756 Simplifying t + -1.333333334 = 2.788866756 Reorder the terms: -1.333333334 + t = 2.788866756 Solving -1.333333334 + t = 2.788866756 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = 2.788866756 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 2.788866756 + 1.333333334 t = 2.788866756 + 1.333333334 Combine like terms: 2.788866756 + 1.333333334 = 4.12220009 t = 4.12220009 Simplifying t = 4.12220009Subproblem 2
t + -1.333333334 = -2.788866756 Simplifying t + -1.333333334 = -2.788866756 Reorder the terms: -1.333333334 + t = -2.788866756 Solving -1.333333334 + t = -2.788866756 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = -2.788866756 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -2.788866756 + 1.333333334 t = -2.788866756 + 1.333333334 Combine like terms: -2.788866756 + 1.333333334 = -1.455533422 t = -1.455533422 Simplifying t = -1.455533422Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.12220009, -1.455533422}Solution
t = {0, 4.12220009, -1.455533422}
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