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Simplifying 15 = 16t2 + 29t + 6 Reorder the terms: 15 = 6 + 29t + 16t2 Solving 15 = 6 + 29t + 16t2 Solving for variable 't'. Combine like terms: 15 + -6 = 9 9 + -29t + -16t2 = 6 + 29t + 16t2 + -6 + -29t + -16t2 Reorder the terms: 9 + -29t + -16t2 = 6 + -6 + 29t + -29t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 9 + -29t + -16t2 = 0 + 29t + -29t + 16t2 + -16t2 9 + -29t + -16t2 = 29t + -29t + 16t2 + -16t2 Combine like terms: 29t + -29t = 0 9 + -29t + -16t2 = 0 + 16t2 + -16t2 9 + -29t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 9 + -29t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5625 + 1.8125t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + 1.8125t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + 1.8125t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + 1.8125t + t2 = 0 + 0.5625 1.8125t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 1.8125t + t2 = 0.5625 The t term is 1.8125t. Take half its coefficient (0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. 1.8125t + 0.8212890625 + t2 = 0.5625 + 0.8212890625 Reorder the terms: 0.8212890625 + 1.8125t + t2 = 0.5625 + 0.8212890625 Combine like terms: 0.5625 + 0.8212890625 = 1.3837890625 0.8212890625 + 1.8125t + t2 = 1.3837890625 Factor a perfect square on the left side: (t + 0.90625)(t + 0.90625) = 1.3837890625 Calculate the square root of the right side: 1.176345639 Break this problem into two subproblems by setting (t + 0.90625) equal to 1.176345639 and -1.176345639.Subproblem 1
t + 0.90625 = 1.176345639 Simplifying t + 0.90625 = 1.176345639 Reorder the terms: 0.90625 + t = 1.176345639 Solving 0.90625 + t = 1.176345639 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = 1.176345639 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = 1.176345639 + -0.90625 t = 1.176345639 + -0.90625 Combine like terms: 1.176345639 + -0.90625 = 0.270095639 t = 0.270095639 Simplifying t = 0.270095639Subproblem 2
t + 0.90625 = -1.176345639 Simplifying t + 0.90625 = -1.176345639 Reorder the terms: 0.90625 + t = -1.176345639 Solving 0.90625 + t = -1.176345639 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + t = -1.176345639 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + t = -1.176345639 + -0.90625 t = -1.176345639 + -0.90625 Combine like terms: -1.176345639 + -0.90625 = -2.082595639 t = -2.082595639 Simplifying t = -2.082595639Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.270095639, -2.082595639}
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