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Simplifying 12 = 16t2 + 24t + 3 Reorder the terms: 12 = 3 + 24t + 16t2 Solving 12 = 3 + 24t + 16t2 Solving for variable 't'. Combine like terms: 12 + -3 = 9 9 + -24t + -16t2 = 3 + 24t + 16t2 + -3 + -24t + -16t2 Reorder the terms: 9 + -24t + -16t2 = 3 + -3 + 24t + -24t + 16t2 + -16t2 Combine like terms: 3 + -3 = 0 9 + -24t + -16t2 = 0 + 24t + -24t + 16t2 + -16t2 9 + -24t + -16t2 = 24t + -24t + 16t2 + -16t2 Combine like terms: 24t + -24t = 0 9 + -24t + -16t2 = 0 + 16t2 + -16t2 9 + -24t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 9 + -24t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5625 + 1.5t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + 1.5t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + 1.5t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + 1.5t + t2 = 0 + 0.5625 1.5t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 1.5t + t2 = 0.5625 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 0.5625 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 0.5625 + 0.5625 Combine like terms: 0.5625 + 0.5625 = 1.125 0.5625 + 1.5t + t2 = 1.125 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (t + 0.75) equal to 1.060660172 and -1.060660172.Subproblem 1
t + 0.75 = 1.060660172 Simplifying t + 0.75 = 1.060660172 Reorder the terms: 0.75 + t = 1.060660172 Solving 0.75 + t = 1.060660172 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 1.060660172 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 1.060660172 + -0.75 t = 1.060660172 + -0.75 Combine like terms: 1.060660172 + -0.75 = 0.310660172 t = 0.310660172 Simplifying t = 0.310660172Subproblem 2
t + 0.75 = -1.060660172 Simplifying t + 0.75 = -1.060660172 Reorder the terms: 0.75 + t = -1.060660172 Solving 0.75 + t = -1.060660172 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -1.060660172 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -1.060660172 + -0.75 t = -1.060660172 + -0.75 Combine like terms: -1.060660172 + -0.75 = -1.810660172 t = -1.810660172 Simplifying t = -1.810660172Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.310660172, -1.810660172}
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