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Simplifying t = -16t2 + 6t + 12 Reorder the terms: t = 12 + 6t + -16t2 Solving t = 12 + 6t + -16t2 Solving for variable 't'. Reorder the terms: -12 + t + -6t + 16t2 = 12 + 6t + -16t2 + -12 + -6t + 16t2 Combine like terms: t + -6t = -5t -12 + -5t + 16t2 = 12 + 6t + -16t2 + -12 + -6t + 16t2 Reorder the terms: -12 + -5t + 16t2 = 12 + -12 + 6t + -6t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -5t + 16t2 = 0 + 6t + -6t + -16t2 + 16t2 -12 + -5t + 16t2 = 6t + -6t + -16t2 + 16t2 Combine like terms: 6t + -6t = 0 -12 + -5t + 16t2 = 0 + -16t2 + 16t2 -12 + -5t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -5t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.75 + -0.3125t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.3125t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.3125t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.3125t + t2 = 0 + 0.75 -0.3125t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.3125t + t2 = 0.75 The t term is -0.3125t. Take half its coefficient (-0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. -0.3125t + 0.0244140625 + t2 = 0.75 + 0.0244140625 Reorder the terms: 0.0244140625 + -0.3125t + t2 = 0.75 + 0.0244140625 Combine like terms: 0.75 + 0.0244140625 = 0.7744140625 0.0244140625 + -0.3125t + t2 = 0.7744140625 Factor a perfect square on the left side: (t + -0.15625)(t + -0.15625) = 0.7744140625 Calculate the square root of the right side: 0.88000799 Break this problem into two subproblems by setting (t + -0.15625) equal to 0.88000799 and -0.88000799.Subproblem 1
t + -0.15625 = 0.88000799 Simplifying t + -0.15625 = 0.88000799 Reorder the terms: -0.15625 + t = 0.88000799 Solving -0.15625 + t = 0.88000799 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + t = 0.88000799 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + t = 0.88000799 + 0.15625 t = 0.88000799 + 0.15625 Combine like terms: 0.88000799 + 0.15625 = 1.03625799 t = 1.03625799 Simplifying t = 1.03625799Subproblem 2
t + -0.15625 = -0.88000799 Simplifying t + -0.15625 = -0.88000799 Reorder the terms: -0.15625 + t = -0.88000799 Solving -0.15625 + t = -0.88000799 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.15625' to each side of the equation. -0.15625 + 0.15625 + t = -0.88000799 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + t = -0.88000799 + 0.15625 t = -0.88000799 + 0.15625 Combine like terms: -0.88000799 + 0.15625 = -0.72375799 t = -0.72375799 Simplifying t = -0.72375799Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.03625799, -0.72375799}
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