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Simplifying (2t2 + 3t) + -1(-9t2 + -8t + 6) = 0 Reorder the terms: (3t + 2t2) + -1(-9t2 + -8t + 6) = 0 Remove parenthesis around (3t + 2t2) 3t + 2t2 + -1(-9t2 + -8t + 6) = 0 Reorder the terms: 3t + 2t2 + -1(6 + -8t + -9t2) = 0 3t + 2t2 + (6 * -1 + -8t * -1 + -9t2 * -1) = 0 3t + 2t2 + (-6 + 8t + 9t2) = 0 Reorder the terms: -6 + 3t + 8t + 2t2 + 9t2 = 0 Combine like terms: 3t + 8t = 11t -6 + 11t + 2t2 + 9t2 = 0 Combine like terms: 2t2 + 9t2 = 11t2 -6 + 11t + 11t2 = 0 Solving -6 + 11t + 11t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.5454545455 + t + t2 = 0 Move the constant term to the right: Add '0.5454545455' to each side of the equation. -0.5454545455 + t + 0.5454545455 + t2 = 0 + 0.5454545455 Reorder the terms: -0.5454545455 + 0.5454545455 + t + t2 = 0 + 0.5454545455 Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000 0.0000000000 + t + t2 = 0 + 0.5454545455 t + t2 = 0 + 0.5454545455 Combine like terms: 0 + 0.5454545455 = 0.5454545455 t + t2 = 0.5454545455 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 0.5454545455 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 0.5454545455 + 0.25 Combine like terms: 0.5454545455 + 0.25 = 0.7954545455 1.25 + t2 = 0.7954545455 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 0.7954545455 Calculate the square root of the right side: 0.891882585 Break this problem into two subproblems by setting (t + 0.5) equal to 0.891882585 and -0.891882585.Subproblem 1
t + 0.5 = 0.891882585 Simplifying t + 0.5 = 0.891882585 Reorder the terms: 0.5 + t = 0.891882585 Solving 0.5 + t = 0.891882585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 0.891882585 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.891882585 + -0.5 t = 0.891882585 + -0.5 Combine like terms: 0.891882585 + -0.5 = 0.391882585 t = 0.391882585 Simplifying t = 0.391882585Subproblem 2
t + 0.5 = -0.891882585 Simplifying t + 0.5 = -0.891882585 Reorder the terms: 0.5 + t = -0.891882585 Solving 0.5 + t = -0.891882585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -0.891882585 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.891882585 + -0.5 t = -0.891882585 + -0.5 Combine like terms: -0.891882585 + -0.5 = -1.391882585 t = -1.391882585 Simplifying t = -1.391882585Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.391882585, -1.391882585}
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