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Simplifying 2t(t + 1) = t3 + -5t Reorder the terms: 2t(1 + t) = t3 + -5t (1 * 2t + t * 2t) = t3 + -5t (2t + 2t2) = t3 + -5t Reorder the terms: 2t + 2t2 = -5t + t3 Solving 2t + 2t2 = -5t + t3 Solving for variable 't'. Reorder the terms: 2t + 5t + 2t2 + -1t3 = -5t + t3 + 5t + -1t3 Combine like terms: 2t + 5t = 7t 7t + 2t2 + -1t3 = -5t + t3 + 5t + -1t3 Reorder the terms: 7t + 2t2 + -1t3 = -5t + 5t + t3 + -1t3 Combine like terms: -5t + 5t = 0 7t + 2t2 + -1t3 = 0 + t3 + -1t3 7t + 2t2 + -1t3 = t3 + -1t3 Combine like terms: t3 + -1t3 = 0 7t + 2t2 + -1t3 = 0 Factor out the Greatest Common Factor (GCF), 't'. t(7 + 2t + -1t2) = 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 '(7 + 2t + -1t2)' equal to zero and attempt to solve: Simplifying 7 + 2t + -1t2 = 0 Solving 7 + 2t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + -2t + t2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -2t + 7 + t2 = 0 + 7 Reorder the terms: -7 + 7 + -2t + t2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -2t + t2 = 0 + 7 -2t + t2 = 0 + 7 Combine like terms: 0 + 7 = 7 -2t + t2 = 7 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 7 + 1 Reorder the terms: 1 + -2t + t2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + -2t + t2 = 8 Factor a perfect square on the left side: (t + -1)(t + -1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (t + -1) equal to 2.828427125 and -2.828427125.Subproblem 1
t + -1 = 2.828427125 Simplifying t + -1 = 2.828427125 Reorder the terms: -1 + t = 2.828427125 Solving -1 + t = 2.828427125 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + t = 2.828427125 + 1 t = 2.828427125 + 1 Combine like terms: 2.828427125 + 1 = 3.828427125 t = 3.828427125 Simplifying t = 3.828427125Subproblem 2
t + -1 = -2.828427125 Simplifying t + -1 = -2.828427125 Reorder the terms: -1 + t = -2.828427125 Solving -1 + t = -2.828427125 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + t = -2.828427125 + 1 t = -2.828427125 + 1 Combine like terms: -2.828427125 + 1 = -1.828427125 t = -1.828427125 Simplifying t = -1.828427125Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.828427125, -1.828427125}Solution
t = {0, 3.828427125, -1.828427125}
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