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Simplifying t = 3 + 14t + -5t2 Solving t = 3 + 14t + -5t2 Solving for variable 't'. Reorder the terms: -3 + t + -14t + 5t2 = 3 + 14t + -5t2 + -3 + -14t + 5t2 Combine like terms: t + -14t = -13t -3 + -13t + 5t2 = 3 + 14t + -5t2 + -3 + -14t + 5t2 Reorder the terms: -3 + -13t + 5t2 = 3 + -3 + 14t + -14t + -5t2 + 5t2 Combine like terms: 3 + -3 = 0 -3 + -13t + 5t2 = 0 + 14t + -14t + -5t2 + 5t2 -3 + -13t + 5t2 = 14t + -14t + -5t2 + 5t2 Combine like terms: 14t + -14t = 0 -3 + -13t + 5t2 = 0 + -5t2 + 5t2 -3 + -13t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -3 + -13t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + -2.6t + t2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -2.6t + 0.6 + t2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -2.6t + t2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -2.6t + t2 = 0 + 0.6 -2.6t + t2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -2.6t + t2 = 0.6 The t term is -2.6t. Take half its coefficient (-1.3). Square it (1.69) and add it to both sides. Add '1.69' to each side of the equation. -2.6t + 1.69 + t2 = 0.6 + 1.69 Reorder the terms: 1.69 + -2.6t + t2 = 0.6 + 1.69 Combine like terms: 0.6 + 1.69 = 2.29 1.69 + -2.6t + t2 = 2.29 Factor a perfect square on the left side: (t + -1.3)(t + -1.3) = 2.29 Calculate the square root of the right side: 1.513274595 Break this problem into two subproblems by setting (t + -1.3) equal to 1.513274595 and -1.513274595.Subproblem 1
t + -1.3 = 1.513274595 Simplifying t + -1.3 = 1.513274595 Reorder the terms: -1.3 + t = 1.513274595 Solving -1.3 + t = 1.513274595 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + t = 1.513274595 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + t = 1.513274595 + 1.3 t = 1.513274595 + 1.3 Combine like terms: 1.513274595 + 1.3 = 2.813274595 t = 2.813274595 Simplifying t = 2.813274595Subproblem 2
t + -1.3 = -1.513274595 Simplifying t + -1.3 = -1.513274595 Reorder the terms: -1.3 + t = -1.513274595 Solving -1.3 + t = -1.513274595 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + t = -1.513274595 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + t = -1.513274595 + 1.3 t = -1.513274595 + 1.3 Combine like terms: -1.513274595 + 1.3 = -0.213274595 t = -0.213274595 Simplifying t = -0.213274595Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.813274595, -0.213274595}
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