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Simplifying -16t2 + 22t = 3 Reorder the terms: 22t + -16t2 = 3 Solving 22t + -16t2 = 3 Solving for variable 't'. Reorder the terms: -3 + 22t + -16t2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 22t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.1875 + -1.375t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -1.375t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -1.375t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -1.375t + t2 = 0 + -0.1875 -1.375t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -1.375t + t2 = -0.1875 The t term is -1.375t. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375t + 0.47265625 + t2 = -0.1875 + 0.47265625 Reorder the terms: 0.47265625 + -1.375t + t2 = -0.1875 + 0.47265625 Combine like terms: -0.1875 + 0.47265625 = 0.28515625 0.47265625 + -1.375t + t2 = 0.28515625 Factor a perfect square on the left side: (t + -0.6875)(t + -0.6875) = 0.28515625 Calculate the square root of the right side: 0.534000234 Break this problem into two subproblems by setting (t + -0.6875) equal to 0.534000234 and -0.534000234.Subproblem 1
t + -0.6875 = 0.534000234 Simplifying t + -0.6875 = 0.534000234 Reorder the terms: -0.6875 + t = 0.534000234 Solving -0.6875 + t = 0.534000234 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = 0.534000234 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = 0.534000234 + 0.6875 t = 0.534000234 + 0.6875 Combine like terms: 0.534000234 + 0.6875 = 1.221500234 t = 1.221500234 Simplifying t = 1.221500234Subproblem 2
t + -0.6875 = -0.534000234 Simplifying t + -0.6875 = -0.534000234 Reorder the terms: -0.6875 + t = -0.534000234 Solving -0.6875 + t = -0.534000234 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + t = -0.534000234 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + t = -0.534000234 + 0.6875 t = -0.534000234 + 0.6875 Combine like terms: -0.534000234 + 0.6875 = 0.153499766 t = 0.153499766 Simplifying t = 0.153499766Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.221500234, 0.153499766}
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