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Simplifying (7t2 + 8t) + -1(-2t2 + -8t + 7) = 0 Reorder the terms: (8t + 7t2) + -1(-2t2 + -8t + 7) = 0 Remove parenthesis around (8t + 7t2) 8t + 7t2 + -1(-2t2 + -8t + 7) = 0 Reorder the terms: 8t + 7t2 + -1(7 + -8t + -2t2) = 0 8t + 7t2 + (7 * -1 + -8t * -1 + -2t2 * -1) = 0 8t + 7t2 + (-7 + 8t + 2t2) = 0 Reorder the terms: -7 + 8t + 8t + 7t2 + 2t2 = 0 Combine like terms: 8t + 8t = 16t -7 + 16t + 7t2 + 2t2 = 0 Combine like terms: 7t2 + 2t2 = 9t2 -7 + 16t + 9t2 = 0 Solving -7 + 16t + 9t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.7777777778 + 1.777777778t + t2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 1.777777778t + 0.7777777778 + t2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 1.777777778t + t2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 1.777777778t + t2 = 0 + 0.7777777778 1.777777778t + t2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 1.777777778t + t2 = 0.7777777778 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 = 0.7777777778 + 0.7901234570 Reorder the terms: 0.7901234570 + 1.777777778t + t2 = 0.7777777778 + 0.7901234570 Combine like terms: 0.7777777778 + 0.7901234570 = 1.5679012348 0.7901234570 + 1.777777778t + t2 = 1.5679012348 Factor a perfect square on the left side: (t + 0.888888889)(t + 0.888888889) = 1.5679012348 Calculate the square root of the right side: 1.25215863 Break this problem into two subproblems by setting (t + 0.888888889) equal to 1.25215863 and -1.25215863.Subproblem 1
t + 0.888888889 = 1.25215863 Simplifying t + 0.888888889 = 1.25215863 Reorder the terms: 0.888888889 + t = 1.25215863 Solving 0.888888889 + t = 1.25215863 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 = 1.25215863 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + t = 1.25215863 + -0.888888889 t = 1.25215863 + -0.888888889 Combine like terms: 1.25215863 + -0.888888889 = 0.363269741 t = 0.363269741 Simplifying t = 0.363269741Subproblem 2
t + 0.888888889 = -1.25215863 Simplifying t + 0.888888889 = -1.25215863 Reorder the terms: 0.888888889 + t = -1.25215863 Solving 0.888888889 + t = -1.25215863 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 = -1.25215863 + -0.888888889 Combine like terms: 0.888888889 + -0.888888889 = 0.000000000 0.000000000 + t = -1.25215863 + -0.888888889 t = -1.25215863 + -0.888888889 Combine like terms: -1.25215863 + -0.888888889 = -2.141047519 t = -2.141047519 Simplifying t = -2.141047519Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.363269741, -2.141047519}
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