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Simplifying 3 + 15t + -5t2 = 11 Solving 3 + 15t + -5t2 = 11 Solving for variable 't'. Reorder the terms: 3 + -11 + 15t + -5t2 = 11 + -11 Combine like terms: 3 + -11 = -8 -8 + 15t + -5t2 = 11 + -11 Combine like terms: 11 + -11 = 0 -8 + 15t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 1.6 + -3t + t2 = 0 Move the constant term to the right: Add '-1.6' to each side of the equation. 1.6 + -3t + -1.6 + t2 = 0 + -1.6 Reorder the terms: 1.6 + -1.6 + -3t + t2 = 0 + -1.6 Combine like terms: 1.6 + -1.6 = 0.0 0.0 + -3t + t2 = 0 + -1.6 -3t + t2 = 0 + -1.6 Combine like terms: 0 + -1.6 = -1.6 -3t + t2 = -1.6 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 = -1.6 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -1.6 + 2.25 Combine like terms: -1.6 + 2.25 = 0.65 2.25 + -3t + t2 = 0.65 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 0.65 Calculate the square root of the right side: 0.806225775 Break this problem into two subproblems by setting (t + -1.5) equal to 0.806225775 and -0.806225775.Subproblem 1
t + -1.5 = 0.806225775 Simplifying t + -1.5 = 0.806225775 Reorder the terms: -1.5 + t = 0.806225775 Solving -1.5 + t = 0.806225775 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 = 0.806225775 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 0.806225775 + 1.5 t = 0.806225775 + 1.5 Combine like terms: 0.806225775 + 1.5 = 2.306225775 t = 2.306225775 Simplifying t = 2.306225775Subproblem 2
t + -1.5 = -0.806225775 Simplifying t + -1.5 = -0.806225775 Reorder the terms: -1.5 + t = -0.806225775 Solving -1.5 + t = -0.806225775 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 = -0.806225775 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -0.806225775 + 1.5 t = -0.806225775 + 1.5 Combine like terms: -0.806225775 + 1.5 = 0.693774225 t = 0.693774225 Simplifying t = 0.693774225Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.306225775, 0.693774225}
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