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Simplifying 23 = 5 + 37t + -16t2 Solving 23 = 5 + 37t + -16t2 Solving for variable 't'. Combine like terms: 23 + -5 = 18 18 + -37t + 16t2 = 5 + 37t + -16t2 + -5 + -37t + 16t2 Reorder the terms: 18 + -37t + 16t2 = 5 + -5 + 37t + -37t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 18 + -37t + 16t2 = 0 + 37t + -37t + -16t2 + 16t2 18 + -37t + 16t2 = 37t + -37t + -16t2 + 16t2 Combine like terms: 37t + -37t = 0 18 + -37t + 16t2 = 0 + -16t2 + 16t2 18 + -37t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 18 + -37t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.125 + -2.3125t + t2 = 0 Move the constant term to the right: Add '-1.125' to each side of the equation. 1.125 + -2.3125t + -1.125 + t2 = 0 + -1.125 Reorder the terms: 1.125 + -1.125 + -2.3125t + t2 = 0 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + -2.3125t + t2 = 0 + -1.125 -2.3125t + t2 = 0 + -1.125 Combine like terms: 0 + -1.125 = -1.125 -2.3125t + t2 = -1.125 The t term is -2.3125t. Take half its coefficient (-1.15625). Square it (1.336914063) and add it to both sides. Add '1.336914063' to each side of the equation. -2.3125t + 1.336914063 + t2 = -1.125 + 1.336914063 Reorder the terms: 1.336914063 + -2.3125t + t2 = -1.125 + 1.336914063 Combine like terms: -1.125 + 1.336914063 = 0.211914063 1.336914063 + -2.3125t + t2 = 0.211914063 Factor a perfect square on the left side: (t + -1.15625)(t + -1.15625) = 0.211914063 Calculate the square root of the right side: 0.460341246 Break this problem into two subproblems by setting (t + -1.15625) equal to 0.460341246 and -0.460341246.Subproblem 1
t + -1.15625 = 0.460341246 Simplifying t + -1.15625 = 0.460341246 Reorder the terms: -1.15625 + t = 0.460341246 Solving -1.15625 + t = 0.460341246 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.15625' to each side of the equation. -1.15625 + 1.15625 + t = 0.460341246 + 1.15625 Combine like terms: -1.15625 + 1.15625 = 0.00000 0.00000 + t = 0.460341246 + 1.15625 t = 0.460341246 + 1.15625 Combine like terms: 0.460341246 + 1.15625 = 1.616591246 t = 1.616591246 Simplifying t = 1.616591246Subproblem 2
t + -1.15625 = -0.460341246 Simplifying t + -1.15625 = -0.460341246 Reorder the terms: -1.15625 + t = -0.460341246 Solving -1.15625 + t = -0.460341246 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.15625' to each side of the equation. -1.15625 + 1.15625 + t = -0.460341246 + 1.15625 Combine like terms: -1.15625 + 1.15625 = 0.00000 0.00000 + t = -0.460341246 + 1.15625 t = -0.460341246 + 1.15625 Combine like terms: -0.460341246 + 1.15625 = 0.695908754 t = 0.695908754 Simplifying t = 0.695908754Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.616591246, 0.695908754}
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