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Simplifying -2t2 + 12t + 9 = 24.5 Reorder the terms: 9 + 12t + -2t2 = 24.5 Solving 9 + 12t + -2t2 = 24.5 Solving for variable 't'. Reorder the terms: 9 + -24.5 + 12t + -2t2 = 24.5 + -24.5 Combine like terms: 9 + -24.5 = -15.5 -15.5 + 12t + -2t2 = 24.5 + -24.5 Combine like terms: 24.5 + -24.5 = 0.0 -15.5 + 12t + -2t2 = 0.0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 7.75 + -6t + t2 = 0.0 Move the constant term to the right: Add '-7.75' to each side of the equation. 7.75 + -6t + -7.75 + t2 = 0.0 + -7.75 Reorder the terms: 7.75 + -7.75 + -6t + t2 = 0.0 + -7.75 Combine like terms: 7.75 + -7.75 = 0.00 0.00 + -6t + t2 = 0.0 + -7.75 -6t + t2 = 0.0 + -7.75 Combine like terms: 0.0 + -7.75 = -7.75 -6t + t2 = -7.75 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -7.75 + 9 Reorder the terms: 9 + -6t + t2 = -7.75 + 9 Combine like terms: -7.75 + 9 = 1.25 9 + -6t + t2 = 1.25 Factor a perfect square on the left side: (t + -3)(t + -3) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (t + -3) equal to 1.118033989 and -1.118033989.Subproblem 1
t + -3 = 1.118033989 Simplifying t + -3 = 1.118033989 Reorder the terms: -3 + t = 1.118033989 Solving -3 + t = 1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + t = 1.118033989 + 3 t = 1.118033989 + 3 Combine like terms: 1.118033989 + 3 = 4.118033989 t = 4.118033989 Simplifying t = 4.118033989Subproblem 2
t + -3 = -1.118033989 Simplifying t + -3 = -1.118033989 Reorder the terms: -3 + t = -1.118033989 Solving -3 + t = -1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + t = -1.118033989 + 3 t = -1.118033989 + 3 Combine like terms: -1.118033989 + 3 = 1.881966011 t = 1.881966011 Simplifying t = 1.881966011Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.118033989, 1.881966011}
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