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Simplifying 0 = 12t(t2 + t + -1) Reorder the terms: 0 = 12t(-1 + t + t2) 0 = (-1 * 12t + t * 12t + t2 * 12t) 0 = (-12t + 12t2 + 12t3) Solving 0 = -12t + 12t2 + 12t3 Solving for variable 't'. Remove the zero: 12t + -12t2 + -12t3 = -12t + 12t2 + 12t3 + 12t + -12t2 + -12t3 Reorder the terms: 12t + -12t2 + -12t3 = -12t + 12t + 12t2 + -12t2 + 12t3 + -12t3 Combine like terms: -12t + 12t = 0 12t + -12t2 + -12t3 = 0 + 12t2 + -12t2 + 12t3 + -12t3 12t + -12t2 + -12t3 = 12t2 + -12t2 + 12t3 + -12t3 Combine like terms: 12t2 + -12t2 = 0 12t + -12t2 + -12t3 = 0 + 12t3 + -12t3 12t + -12t2 + -12t3 = 12t3 + -12t3 Combine like terms: 12t3 + -12t3 = 0 12t + -12t2 + -12t3 = 0 Factor out the Greatest Common Factor (GCF), '12t'. 12t(1 + -1t + -1t2) = 0 Ignore the factor 12.Subproblem 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 '(1 + -1t + -1t2)' equal to zero and attempt to solve: Simplifying 1 + -1t + -1t2 = 0 Solving 1 + -1t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + t + t2 = 0 + 1 t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 t + t2 = 1 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 = 1 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 1.25 + t2 = 1.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (t + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
t + 0.5 = 1.118033989 Simplifying t + 0.5 = 1.118033989 Reorder the terms: 0.5 + t = 1.118033989 Solving 0.5 + t = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.118033989 + -0.5 t = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 t = 0.618033989 Simplifying t = 0.618033989Subproblem 2
t + 0.5 = -1.118033989 Simplifying t + 0.5 = -1.118033989 Reorder the terms: 0.5 + t = -1.118033989 Solving 0.5 + t = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.118033989 + -0.5 t = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 t = -1.618033989 Simplifying t = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.618033989, -1.618033989}Solution
t = {0, 0.618033989, -1.618033989}
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