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Simplifying 9 = 3 + 15t + -5t2 Solving 9 = 3 + 15t + -5t2 Solving for variable 't'. Combine like terms: 9 + -3 = 6 6 + -15t + 5t2 = 3 + 15t + -5t2 + -3 + -15t + 5t2 Reorder the terms: 6 + -15t + 5t2 = 3 + -3 + 15t + -15t + -5t2 + 5t2 Combine like terms: 3 + -3 = 0 6 + -15t + 5t2 = 0 + 15t + -15t + -5t2 + 5t2 6 + -15t + 5t2 = 15t + -15t + -5t2 + 5t2 Combine like terms: 15t + -15t = 0 6 + -15t + 5t2 = 0 + -5t2 + 5t2 6 + -15t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 6 + -15t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1.2 + -3t + t2 = 0 Move the constant term to the right: Add '-1.2' to each side of the equation. 1.2 + -3t + -1.2 + t2 = 0 + -1.2 Reorder the terms: 1.2 + -1.2 + -3t + t2 = 0 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + -3t + t2 = 0 + -1.2 -3t + t2 = 0 + -1.2 Combine like terms: 0 + -1.2 = -1.2 -3t + t2 = -1.2 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -1.2 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -1.2 + 2.25 Combine like terms: -1.2 + 2.25 = 1.05 2.25 + -3t + t2 = 1.05 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1.05 Calculate the square root of the right side: 1.024695077 Break this problem into two subproblems by setting (t + -1.5) equal to 1.024695077 and -1.024695077.Subproblem 1
t + -1.5 = 1.024695077 Simplifying t + -1.5 = 1.024695077 Reorder the terms: -1.5 + t = 1.024695077 Solving -1.5 + t = 1.024695077 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.024695077 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.024695077 + 1.5 t = 1.024695077 + 1.5 Combine like terms: 1.024695077 + 1.5 = 2.524695077 t = 2.524695077 Simplifying t = 2.524695077Subproblem 2
t + -1.5 = -1.024695077 Simplifying t + -1.5 = -1.024695077 Reorder the terms: -1.5 + t = -1.024695077 Solving -1.5 + t = -1.024695077 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.024695077 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.024695077 + 1.5 t = -1.024695077 + 1.5 Combine like terms: -1.024695077 + 1.5 = 0.475304923 t = 0.475304923 Simplifying t = 0.475304923Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.524695077, 0.475304923}
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