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Simplifying 0 = -81t2 + -144t + -13 Reorder the terms: 0 = -13 + -144t + -81t2 Solving 0 = -13 + -144t + -81t2 Solving for variable 't'. Combine like terms: 0 + 13 = 13 13 + 144t + 81t2 = -13 + -144t + -81t2 + 13 + 144t + 81t2 Reorder the terms: 13 + 144t + 81t2 = -13 + 13 + -144t + 144t + -81t2 + 81t2 Combine like terms: -13 + 13 = 0 13 + 144t + 81t2 = 0 + -144t + 144t + -81t2 + 81t2 13 + 144t + 81t2 = -144t + 144t + -81t2 + 81t2 Combine like terms: -144t + 144t = 0 13 + 144t + 81t2 = 0 + -81t2 + 81t2 13 + 144t + 81t2 = -81t2 + 81t2 Combine like terms: -81t2 + 81t2 = 0 13 + 144t + 81t2 = 0 Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. 0.1604938272 + 1.777777778t + t2 = 0 Move the constant term to the right: Add '-0.1604938272' to each side of the equation. 0.1604938272 + 1.777777778t + -0.1604938272 + t2 = 0 + -0.1604938272 Reorder the terms: 0.1604938272 + -0.1604938272 + 1.777777778t + t2 = 0 + -0.1604938272 Combine like terms: 0.1604938272 + -0.1604938272 = 0.0000000000 0.0000000000 + 1.777777778t + t2 = 0 + -0.1604938272 1.777777778t + t2 = 0 + -0.1604938272 Combine like terms: 0 + -0.1604938272 = -0.1604938272 1.777777778t + t2 = -0.1604938272 The t term is 1.777777778t. Take half its coefficient (0.888888889). Square it (0.7901234570) and add it to both sides. Add '0.7901234570' to each side of the equation. 1.777777778t + 0.7901234570 + t2 = -0.1604938272 + 0.7901234570 Reorder the terms: 0.7901234570 + 1.777777778t + t2 = -0.1604938272 + 0.7901234570 Combine like terms: -0.1604938272 + 0.7901234570 = 0.6296296298 0.7901234570 + 1.777777778t + t2 = 0.6296296298 Factor a perfect square on the left side: (t + 0.888888889)(t + 0.888888889) = 0.6296296298 Calculate the square root of the right side: 0.793492048 Break this problem into two subproblems by setting (t + 0.888888889) equal to 0.793492048 and -0.793492048.Subproblem 1
t + 0.888888889 = 0.793492048 Simplifying t + 0.888888889 = 0.793492048 Reorder the terms: 0.888888889 + t = 0.793492048 Solving 0.888888889 + t = 0.793492048 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + t = 0.793492048 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + t = 0.793492048 + -0.888888889 t = 0.793492048 + -0.888888889 Combine like terms: 0.793492048 + -0.888888889 = -0.095396841 t = -0.095396841 Simplifying t = -0.095396841Subproblem 2
t + 0.888888889 = -0.793492048 Simplifying t + 0.888888889 = -0.793492048 Reorder the terms: 0.888888889 + t = -0.793492048 Solving 0.888888889 + t = -0.793492048 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.888888889' to each side of the equation. 0.888888889 + -0.888888889 + t = -0.793492048 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + t = -0.793492048 + -0.888888889 t = -0.793492048 + -0.888888889 Combine like terms: -0.793492048 + -0.888888889 = -1.682380937 t = -1.682380937 Simplifying t = -1.682380937Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.095396841, -1.682380937}
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