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Simplifying (2t2 + 3t) + -1(-6t2 + -9t + 6) = 0 Reorder the terms: (3t + 2t2) + -1(-6t2 + -9t + 6) = 0 Remove parenthesis around (3t + 2t2) 3t + 2t2 + -1(-6t2 + -9t + 6) = 0 Reorder the terms: 3t + 2t2 + -1(6 + -9t + -6t2) = 0 3t + 2t2 + (6 * -1 + -9t * -1 + -6t2 * -1) = 0 3t + 2t2 + (-6 + 9t + 6t2) = 0 Reorder the terms: -6 + 3t + 9t + 2t2 + 6t2 = 0 Combine like terms: 3t + 9t = 12t -6 + 12t + 2t2 + 6t2 = 0 Combine like terms: 2t2 + 6t2 = 8t2 -6 + 12t + 8t2 = 0 Solving -6 + 12t + 8t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 6t + 4t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 6t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + 6t + 4t2 = 0 Solving -3 + 6t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 1.5t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.5t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.5t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.5t + t2 = 0 + 0.75 1.5t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.5t + t2 = 0.75 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + 1.5t + t2 = 1.3125 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (t + 0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
t + 0.75 = 1.145643924 Simplifying t + 0.75 = 1.145643924 Reorder the terms: 0.75 + t = 1.145643924 Solving 0.75 + t = 1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 1.145643924 + -0.75 t = 1.145643924 + -0.75 Combine like terms: 1.145643924 + -0.75 = 0.395643924 t = 0.395643924 Simplifying t = 0.395643924Subproblem 2
t + 0.75 = -1.145643924 Simplifying t + 0.75 = -1.145643924 Reorder the terms: 0.75 + t = -1.145643924 Solving 0.75 + t = -1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -1.145643924 + -0.75 t = -1.145643924 + -0.75 Combine like terms: -1.145643924 + -0.75 = -1.895643924 t = -1.895643924 Simplifying t = -1.895643924Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.395643924, -1.895643924}Solution
t = {0.395643924, -1.895643924}
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