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Simplifying 24t2 + 24t + -35 = 0 Reorder the terms: -35 + 24t + 24t2 = 0 Solving -35 + 24t + 24t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -1.458333333 + t + t2 = 0 Move the constant term to the right: Add '1.458333333' to each side of the equation. -1.458333333 + t + 1.458333333 + t2 = 0 + 1.458333333 Reorder the terms: -1.458333333 + 1.458333333 + t + t2 = 0 + 1.458333333 Combine like terms: -1.458333333 + 1.458333333 = 0.000000000 0.000000000 + t + t2 = 0 + 1.458333333 t + t2 = 0 + 1.458333333 Combine like terms: 0 + 1.458333333 = 1.458333333 t + t2 = 1.458333333 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 = 1.458333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 1.458333333 + 0.25 Combine like terms: 1.458333333 + 0.25 = 1.708333333 1.25 + t2 = 1.708333333 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 1.708333333 Calculate the square root of the right side: 1.307032262 Break this problem into two subproblems by setting (t + 0.5) equal to 1.307032262 and -1.307032262.Subproblem 1
t + 0.5 = 1.307032262 Simplifying t + 0.5 = 1.307032262 Reorder the terms: 0.5 + t = 1.307032262 Solving 0.5 + t = 1.307032262 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 = 1.307032262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.307032262 + -0.5 t = 1.307032262 + -0.5 Combine like terms: 1.307032262 + -0.5 = 0.807032262 t = 0.807032262 Simplifying t = 0.807032262Subproblem 2
t + 0.5 = -1.307032262 Simplifying t + 0.5 = -1.307032262 Reorder the terms: 0.5 + t = -1.307032262 Solving 0.5 + t = -1.307032262 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 = -1.307032262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.307032262 + -0.5 t = -1.307032262 + -0.5 Combine like terms: -1.307032262 + -0.5 = -1.807032262 t = -1.807032262 Simplifying t = -1.807032262Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.807032262, -1.807032262}
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