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Simplifying t2 + 30 = 2(10 + -8t) Reorder the terms: 30 + t2 = 2(10 + -8t) 30 + t2 = (10 * 2 + -8t * 2) 30 + t2 = (20 + -16t) Solving 30 + t2 = 20 + -16t Solving for variable 't'. Reorder the terms: 30 + -20 + 16t + t2 = 20 + -16t + -20 + 16t Combine like terms: 30 + -20 = 10 10 + 16t + t2 = 20 + -16t + -20 + 16t Reorder the terms: 10 + 16t + t2 = 20 + -20 + -16t + 16t Combine like terms: 20 + -20 = 0 10 + 16t + t2 = 0 + -16t + 16t 10 + 16t + t2 = -16t + 16t Combine like terms: -16t + 16t = 0 10 + 16t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-10' to each side of the equation. 10 + 16t + -10 + t2 = 0 + -10 Reorder the terms: 10 + -10 + 16t + t2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 16t + t2 = 0 + -10 16t + t2 = 0 + -10 Combine like terms: 0 + -10 = -10 16t + t2 = -10 The t term is 16t. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16t + 64 + t2 = -10 + 64 Reorder the terms: 64 + 16t + t2 = -10 + 64 Combine like terms: -10 + 64 = 54 64 + 16t + t2 = 54 Factor a perfect square on the left side: (t + 8)(t + 8) = 54 Calculate the square root of the right side: 7.348469228 Break this problem into two subproblems by setting (t + 8) equal to 7.348469228 and -7.348469228.Subproblem 1
t + 8 = 7.348469228 Simplifying t + 8 = 7.348469228 Reorder the terms: 8 + t = 7.348469228 Solving 8 + t = 7.348469228 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + t = 7.348469228 + -8 Combine like terms: 8 + -8 = 0 0 + t = 7.348469228 + -8 t = 7.348469228 + -8 Combine like terms: 7.348469228 + -8 = -0.651530772 t = -0.651530772 Simplifying t = -0.651530772Subproblem 2
t + 8 = -7.348469228 Simplifying t + 8 = -7.348469228 Reorder the terms: 8 + t = -7.348469228 Solving 8 + t = -7.348469228 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + t = -7.348469228 + -8 Combine like terms: 8 + -8 = 0 0 + t = -7.348469228 + -8 t = -7.348469228 + -8 Combine like terms: -7.348469228 + -8 = -15.348469228 t = -15.348469228 Simplifying t = -15.348469228Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.651530772, -15.348469228}
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