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Simplifying (2t2 + 3t + -4) + (8t + 1) = 0 Reorder the terms: (-4 + 3t + 2t2) + (8t + 1) = 0 Remove parenthesis around (-4 + 3t + 2t2) -4 + 3t + 2t2 + (8t + 1) = 0 Reorder the terms: -4 + 3t + 2t2 + (1 + 8t) = 0 Remove parenthesis around (1 + 8t) -4 + 3t + 2t2 + 1 + 8t = 0 Reorder the terms: -4 + 1 + 3t + 8t + 2t2 = 0 Combine like terms: -4 + 1 = -3 -3 + 3t + 8t + 2t2 = 0 Combine like terms: 3t + 8t = 11t -3 + 11t + 2t2 = 0 Solving -3 + 11t + 2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 5.5t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 5.5t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 5.5t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 5.5t + t2 = 0 + 1.5 5.5t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 5.5t + t2 = 1.5 The t term is 5.5t. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5t + 7.5625 + t2 = 1.5 + 7.5625 Reorder the terms: 7.5625 + 5.5t + t2 = 1.5 + 7.5625 Combine like terms: 1.5 + 7.5625 = 9.0625 7.5625 + 5.5t + t2 = 9.0625 Factor a perfect square on the left side: (t + 2.75)(t + 2.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (t + 2.75) equal to 3.010398645 and -3.010398645.Subproblem 1
t + 2.75 = 3.010398645 Simplifying t + 2.75 = 3.010398645 Reorder the terms: 2.75 + t = 3.010398645 Solving 2.75 + t = 3.010398645 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + t = 3.010398645 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + t = 3.010398645 + -2.75 t = 3.010398645 + -2.75 Combine like terms: 3.010398645 + -2.75 = 0.260398645 t = 0.260398645 Simplifying t = 0.260398645Subproblem 2
t + 2.75 = -3.010398645 Simplifying t + 2.75 = -3.010398645 Reorder the terms: 2.75 + t = -3.010398645 Solving 2.75 + t = -3.010398645 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + t = -3.010398645 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + t = -3.010398645 + -2.75 t = -3.010398645 + -2.75 Combine like terms: -3.010398645 + -2.75 = -5.760398645 t = -5.760398645 Simplifying t = -5.760398645Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.260398645, -5.760398645}
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