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Simplifying 193 = 16t2 + 32t Reorder the terms: 193 = 32t + 16t2 Solving 193 = 32t + 16t2 Solving for variable 't'. Reorder the terms: 193 + -32t + -16t2 = 32t + -32t + 16t2 + -16t2 Combine like terms: 32t + -32t = 0 193 + -32t + -16t2 = 0 + 16t2 + -16t2 193 + -32t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 193 + -32t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -12.0625 + 2t + t2 = 0 Move the constant term to the right: Add '12.0625' to each side of the equation. -12.0625 + 2t + 12.0625 + t2 = 0 + 12.0625 Reorder the terms: -12.0625 + 12.0625 + 2t + t2 = 0 + 12.0625 Combine like terms: -12.0625 + 12.0625 = 0.0000 0.0000 + 2t + t2 = 0 + 12.0625 2t + t2 = 0 + 12.0625 Combine like terms: 0 + 12.0625 = 12.0625 2t + t2 = 12.0625 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 12.0625 + 1 Reorder the terms: 1 + 2t + t2 = 12.0625 + 1 Combine like terms: 12.0625 + 1 = 13.0625 1 + 2t + t2 = 13.0625 Factor a perfect square on the left side: (t + 1)(t + 1) = 13.0625 Calculate the square root of the right side: 3.614208074 Break this problem into two subproblems by setting (t + 1) equal to 3.614208074 and -3.614208074.Subproblem 1
t + 1 = 3.614208074 Simplifying t + 1 = 3.614208074 Reorder the terms: 1 + t = 3.614208074 Solving 1 + t = 3.614208074 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 3.614208074 + -1 Combine like terms: 1 + -1 = 0 0 + t = 3.614208074 + -1 t = 3.614208074 + -1 Combine like terms: 3.614208074 + -1 = 2.614208074 t = 2.614208074 Simplifying t = 2.614208074Subproblem 2
t + 1 = -3.614208074 Simplifying t + 1 = -3.614208074 Reorder the terms: 1 + t = -3.614208074 Solving 1 + t = -3.614208074 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -3.614208074 + -1 Combine like terms: 1 + -1 = 0 0 + t = -3.614208074 + -1 t = -3.614208074 + -1 Combine like terms: -3.614208074 + -1 = -4.614208074 t = -4.614208074 Simplifying t = -4.614208074Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.614208074, -4.614208074}
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