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Simplifying 16t2 = 23t + 7 Reorder the terms: 16t2 = 7 + 23t Solving 16t2 = 7 + 23t Solving for variable 't'. Reorder the terms: -7 + -23t + 16t2 = 7 + 23t + -7 + -23t Reorder the terms: -7 + -23t + 16t2 = 7 + -7 + 23t + -23t Combine like terms: 7 + -7 = 0 -7 + -23t + 16t2 = 0 + 23t + -23t -7 + -23t + 16t2 = 23t + -23t Combine like terms: 23t + -23t = 0 -7 + -23t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.4375 + -1.4375t + t2 = 0 Move the constant term to the right: Add '0.4375' to each side of the equation. -0.4375 + -1.4375t + 0.4375 + t2 = 0 + 0.4375 Reorder the terms: -0.4375 + 0.4375 + -1.4375t + t2 = 0 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + -1.4375t + t2 = 0 + 0.4375 -1.4375t + t2 = 0 + 0.4375 Combine like terms: 0 + 0.4375 = 0.4375 -1.4375t + t2 = 0.4375 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 = 0.4375 + 0.5166015625 Reorder the terms: 0.5166015625 + -1.4375t + t2 = 0.4375 + 0.5166015625 Combine like terms: 0.4375 + 0.5166015625 = 0.9541015625 0.5166015625 + -1.4375t + t2 = 0.9541015625 Factor a perfect square on the left side: (t + -0.71875)(t + -0.71875) = 0.9541015625 Calculate the square root of the right side: 0.976781226 Break this problem into two subproblems by setting (t + -0.71875) equal to 0.976781226 and -0.976781226.Subproblem 1
t + -0.71875 = 0.976781226 Simplifying t + -0.71875 = 0.976781226 Reorder the terms: -0.71875 + t = 0.976781226 Solving -0.71875 + t = 0.976781226 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 = 0.976781226 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = 0.976781226 + 0.71875 t = 0.976781226 + 0.71875 Combine like terms: 0.976781226 + 0.71875 = 1.695531226 t = 1.695531226 Simplifying t = 1.695531226Subproblem 2
t + -0.71875 = -0.976781226 Simplifying t + -0.71875 = -0.976781226 Reorder the terms: -0.71875 + t = -0.976781226 Solving -0.71875 + t = -0.976781226 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 = -0.976781226 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + t = -0.976781226 + 0.71875 t = -0.976781226 + 0.71875 Combine like terms: -0.976781226 + 0.71875 = -0.258031226 t = -0.258031226 Simplifying t = -0.258031226Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.695531226, -0.258031226}
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