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Simplifying 9t3 + 15t2 + t + -6 = (t + 3)(t + -2) + -3t Reorder the terms: -6 + t + 15t2 + 9t3 = (t + 3)(t + -2) + -3t Reorder the terms: -6 + t + 15t2 + 9t3 = (3 + t)(t + -2) + -3t Reorder the terms: -6 + t + 15t2 + 9t3 = (3 + t)(-2 + t) + -3t Multiply (3 + t) * (-2 + t) -6 + t + 15t2 + 9t3 = (3(-2 + t) + t(-2 + t)) + -3t -6 + t + 15t2 + 9t3 = ((-2 * 3 + t * 3) + t(-2 + t)) + -3t -6 + t + 15t2 + 9t3 = ((-6 + 3t) + t(-2 + t)) + -3t -6 + t + 15t2 + 9t3 = (-6 + 3t + (-2 * t + t * t)) + -3t -6 + t + 15t2 + 9t3 = (-6 + 3t + (-2t + t2)) + -3t Combine like terms: 3t + -2t = 1t -6 + t + 15t2 + 9t3 = (-6 + 1t + t2) + -3t Reorder the terms: -6 + t + 15t2 + 9t3 = -6 + 1t + -3t + t2 Combine like terms: 1t + -3t = -2t -6 + t + 15t2 + 9t3 = -6 + -2t + t2 Add '6' to each side of the equation. -6 + t + 15t2 + 6 + 9t3 = -6 + -2t + 6 + t2 Reorder the terms: -6 + 6 + t + 15t2 + 9t3 = -6 + -2t + 6 + t2 Combine like terms: -6 + 6 = 0 0 + t + 15t2 + 9t3 = -6 + -2t + 6 + t2 t + 15t2 + 9t3 = -6 + -2t + 6 + t2 Reorder the terms: t + 15t2 + 9t3 = -6 + 6 + -2t + t2 Combine like terms: -6 + 6 = 0 t + 15t2 + 9t3 = 0 + -2t + t2 t + 15t2 + 9t3 = -2t + t2 Solving t + 15t2 + 9t3 = -2t + t2 Solving for variable 't'. Reorder the terms: t + 2t + 15t2 + -1t2 + 9t3 = -2t + t2 + 2t + -1t2 Combine like terms: t + 2t = 3t 3t + 15t2 + -1t2 + 9t3 = -2t + t2 + 2t + -1t2 Combine like terms: 15t2 + -1t2 = 14t2 3t + 14t2 + 9t3 = -2t + t2 + 2t + -1t2 Reorder the terms: 3t + 14t2 + 9t3 = -2t + 2t + t2 + -1t2 Combine like terms: -2t + 2t = 0 3t + 14t2 + 9t3 = 0 + t2 + -1t2 3t + 14t2 + 9t3 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 3t + 14t2 + 9t3 = 0 Factor out the Greatest Common Factor (GCF), 't'. t(3 + 14t + 9t2) = 0Subproblem 1
Set the factor 't' equal to zero and attempt to solve: Simplifying t = 0 Solving t = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t = 0Subproblem 2
Set the factor '(3 + 14t + 9t2)' equal to zero and attempt to solve: Simplifying 3 + 14t + 9t2 = 0 Solving 3 + 14t + 9t2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.3333333333 + 1.555555556t + t2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 1.555555556t + -0.3333333333 + t2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 1.555555556t + t2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 1.555555556t + t2 = 0 + -0.3333333333 1.555555556t + t2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 1.555555556t + t2 = -0.3333333333 The t term is 1.555555556t. Take half its coefficient (0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. 1.555555556t + 0.6049382720 + t2 = -0.3333333333 + 0.6049382720 Reorder the terms: 0.6049382720 + 1.555555556t + t2 = -0.3333333333 + 0.6049382720 Combine like terms: -0.3333333333 + 0.6049382720 = 0.2716049387 0.6049382720 + 1.555555556t + t2 = 0.2716049387 Factor a perfect square on the left side: (t + 0.777777778)(t + 0.777777778) = 0.2716049387 Calculate the square root of the right side: 0.521157307 Break this problem into two subproblems by setting (t + 0.777777778) equal to 0.521157307 and -0.521157307.Subproblem 1
t + 0.777777778 = 0.521157307 Simplifying t + 0.777777778 = 0.521157307 Reorder the terms: 0.777777778 + t = 0.521157307 Solving 0.777777778 + t = 0.521157307 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + t = 0.521157307 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + t = 0.521157307 + -0.777777778 t = 0.521157307 + -0.777777778 Combine like terms: 0.521157307 + -0.777777778 = -0.256620471 t = -0.256620471 Simplifying t = -0.256620471Subproblem 2
t + 0.777777778 = -0.521157307 Simplifying t + 0.777777778 = -0.521157307 Reorder the terms: 0.777777778 + t = -0.521157307 Solving 0.777777778 + t = -0.521157307 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + t = -0.521157307 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + t = -0.521157307 + -0.777777778 t = -0.521157307 + -0.777777778 Combine like terms: -0.521157307 + -0.777777778 = -1.298935085 t = -1.298935085 Simplifying t = -1.298935085Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.256620471, -1.298935085}Solution
t = {0, -0.256620471, -1.298935085}
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