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Simplifying -16t2 + 31t + 3 = 0 Reorder the terms: 3 + 31t + -16t2 = 0 Solving 3 + 31t + -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'. -0.1875 + -1.9375t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -1.9375t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -1.9375t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -1.9375t + t2 = 0 + 0.1875 -1.9375t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -1.9375t + t2 = 0.1875 The t term is -1.9375t. Take half its coefficient (-0.96875). Square it (0.9384765625) and add it to both sides. Add '0.9384765625' to each side of the equation. -1.9375t + 0.9384765625 + t2 = 0.1875 + 0.9384765625 Reorder the terms: 0.9384765625 + -1.9375t + t2 = 0.1875 + 0.9384765625 Combine like terms: 0.1875 + 0.9384765625 = 1.1259765625 0.9384765625 + -1.9375t + t2 = 1.1259765625 Factor a perfect square on the left side: (t + -0.96875)(t + -0.96875) = 1.1259765625 Calculate the square root of the right side: 1.061120428 Break this problem into two subproblems by setting (t + -0.96875) equal to 1.061120428 and -1.061120428.Subproblem 1
t + -0.96875 = 1.061120428 Simplifying t + -0.96875 = 1.061120428 Reorder the terms: -0.96875 + t = 1.061120428 Solving -0.96875 + t = 1.061120428 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = 1.061120428 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.061120428 + 0.96875 t = 1.061120428 + 0.96875 Combine like terms: 1.061120428 + 0.96875 = 2.029870428 t = 2.029870428 Simplifying t = 2.029870428Subproblem 2
t + -0.96875 = -1.061120428 Simplifying t + -0.96875 = -1.061120428 Reorder the terms: -0.96875 + t = -1.061120428 Solving -0.96875 + t = -1.061120428 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = -1.061120428 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.061120428 + 0.96875 t = -1.061120428 + 0.96875 Combine like terms: -1.061120428 + 0.96875 = -0.092370428 t = -0.092370428 Simplifying t = -0.092370428Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.029870428, -0.092370428}
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