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Simplifying 130 = 5.7t2 + 14t + 32 Reorder the terms: 130 = 32 + 14t + 5.7t2 Solving 130 = 32 + 14t + 5.7t2 Solving for variable 't'. Combine like terms: 130 + -32 = 98 98 + -14t + -5.7t2 = 32 + 14t + 5.7t2 + -32 + -14t + -5.7t2 Reorder the terms: 98 + -14t + -5.7t2 = 32 + -32 + 14t + -14t + 5.7t2 + -5.7t2 Combine like terms: 32 + -32 = 0 98 + -14t + -5.7t2 = 0 + 14t + -14t + 5.7t2 + -5.7t2 98 + -14t + -5.7t2 = 14t + -14t + 5.7t2 + -5.7t2 Combine like terms: 14t + -14t = 0 98 + -14t + -5.7t2 = 0 + 5.7t2 + -5.7t2 98 + -14t + -5.7t2 = 5.7t2 + -5.7t2 Combine like terms: 5.7t2 + -5.7t2 = 0.0 98 + -14t + -5.7t2 = 0.0 Begin completing the square. Divide all terms by -5.7 the coefficient of the squared term: Divide each side by '-5.7'. -17.19298246 + 2.456140351t + t2 = 0 Move the constant term to the right: Add '17.19298246' to each side of the equation. -17.19298246 + 2.456140351t + 17.19298246 + t2 = 0 + 17.19298246 Reorder the terms: -17.19298246 + 17.19298246 + 2.456140351t + t2 = 0 + 17.19298246 Combine like terms: -17.19298246 + 17.19298246 = 0.00000000 0.00000000 + 2.456140351t + t2 = 0 + 17.19298246 2.456140351t + t2 = 0 + 17.19298246 Combine like terms: 0 + 17.19298246 = 17.19298246 2.456140351t + t2 = 17.19298246 The t term is 2.456140351t. Take half its coefficient (1.228070176). Square it (1.508156357) and add it to both sides. Add '1.508156357' to each side of the equation. 2.456140351t + 1.508156357 + t2 = 17.19298246 + 1.508156357 Reorder the terms: 1.508156357 + 2.456140351t + t2 = 17.19298246 + 1.508156357 Combine like terms: 17.19298246 + 1.508156357 = 18.701138817 1.508156357 + 2.456140351t + t2 = 18.701138817 Factor a perfect square on the left side: (t + 1.228070176)(t + 1.228070176) = 18.701138817 Calculate the square root of the right side: 4.324481335 Break this problem into two subproblems by setting (t + 1.228070176) equal to 4.324481335 and -4.324481335.Subproblem 1
t + 1.228070176 = 4.324481335 Simplifying t + 1.228070176 = 4.324481335 Reorder the terms: 1.228070176 + t = 4.324481335 Solving 1.228070176 + t = 4.324481335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.228070176' to each side of the equation. 1.228070176 + -1.228070176 + t = 4.324481335 + -1.228070176 Combine like terms: 1.228070176 + -1.228070176 = 0.000000000 0.000000000 + t = 4.324481335 + -1.228070176 t = 4.324481335 + -1.228070176 Combine like terms: 4.324481335 + -1.228070176 = 3.096411159 t = 3.096411159 Simplifying t = 3.096411159Subproblem 2
t + 1.228070176 = -4.324481335 Simplifying t + 1.228070176 = -4.324481335 Reorder the terms: 1.228070176 + t = -4.324481335 Solving 1.228070176 + t = -4.324481335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.228070176' to each side of the equation. 1.228070176 + -1.228070176 + t = -4.324481335 + -1.228070176 Combine like terms: 1.228070176 + -1.228070176 = 0.000000000 0.000000000 + t = -4.324481335 + -1.228070176 t = -4.324481335 + -1.228070176 Combine like terms: -4.324481335 + -1.228070176 = -5.552551511 t = -5.552551511 Simplifying t = -5.552551511Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.096411159, -5.552551511}
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