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Simplifying -16t2 + -23t + 375 = 0 Reorder the terms: 375 + -23t + -16t2 = 0 Solving 375 + -23t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -23.4375 + 1.4375t + t2 = 0 Move the constant term to the right: Add '23.4375' to each side of the equation. -23.4375 + 1.4375t + 23.4375 + t2 = 0 + 23.4375 Reorder the terms: -23.4375 + 23.4375 + 1.4375t + t2 = 0 + 23.4375 Combine like terms: -23.4375 + 23.4375 = 0.0000 0.0000 + 1.4375t + t2 = 0 + 23.4375 1.4375t + t2 = 0 + 23.4375 Combine like terms: 0 + 23.4375 = 23.4375 1.4375t + t2 = 23.4375 The t term is 1.4375t. Take half its coefficient (0.71875). Square it (0.5166015625) and add it to both sides. Add '0.5166015625' to each side of the equation. 1.4375t + 0.5166015625 + t2 = 23.4375 + 0.5166015625 Reorder the terms: 0.5166015625 + 1.4375t + t2 = 23.4375 + 0.5166015625 Combine like terms: 23.4375 + 0.5166015625 = 23.9541015625 0.5166015625 + 1.4375t + t2 = 23.9541015625 Factor a perfect square on the left side: (t + 0.71875)(t + 0.71875) = 23.9541015625 Calculate the square root of the right side: 4.894292754 Break this problem into two subproblems by setting (t + 0.71875) equal to 4.894292754 and -4.894292754.Subproblem 1
t + 0.71875 = 4.894292754 Simplifying t + 0.71875 = 4.894292754 Reorder the terms: 0.71875 + t = 4.894292754 Solving 0.71875 + t = 4.894292754 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.71875' to each side of the equation. 0.71875 + -0.71875 + t = 4.894292754 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + t = 4.894292754 + -0.71875 t = 4.894292754 + -0.71875 Combine like terms: 4.894292754 + -0.71875 = 4.175542754 t = 4.175542754 Simplifying t = 4.175542754Subproblem 2
t + 0.71875 = -4.894292754 Simplifying t + 0.71875 = -4.894292754 Reorder the terms: 0.71875 + t = -4.894292754 Solving 0.71875 + t = -4.894292754 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.71875' to each side of the equation. 0.71875 + -0.71875 + t = -4.894292754 + -0.71875 Combine like terms: 0.71875 + -0.71875 = 0.00000 0.00000 + t = -4.894292754 + -0.71875 t = -4.894292754 + -0.71875 Combine like terms: -4.894292754 + -0.71875 = -5.613042754 t = -5.613042754 Simplifying t = -5.613042754Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.175542754, -5.613042754}
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