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Simplifying 81 = 9t2 + 54t Reorder the terms: 81 = 54t + 9t2 Solving 81 = 54t + 9t2 Solving for variable 't'. Reorder the terms: 81 + -54t + -9t2 = 54t + -54t + 9t2 + -9t2 Combine like terms: 54t + -54t = 0 81 + -54t + -9t2 = 0 + 9t2 + -9t2 81 + -54t + -9t2 = 9t2 + -9t2 Combine like terms: 9t2 + -9t2 = 0 81 + -54t + -9t2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(9 + -6t + -1t2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(9 + -6t + -1t2)' equal to zero and attempt to solve: Simplifying 9 + -6t + -1t2 = 0 Solving 9 + -6t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -9 + 6t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 6t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + 6t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 6t + t2 = 0 + 9 6t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 6t + t2 = 9 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 = 9 + 9 Reorder the terms: 9 + 6t + t2 = 9 + 9 Combine like terms: 9 + 9 = 18 9 + 6t + t2 = 18 Factor a perfect square on the left side: (t + 3)(t + 3) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (t + 3) equal to 4.242640687 and -4.242640687.Subproblem 1
t + 3 = 4.242640687 Simplifying t + 3 = 4.242640687 Reorder the terms: 3 + t = 4.242640687 Solving 3 + t = 4.242640687 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 = 4.242640687 + -3 Combine like terms: 3 + -3 = 0 0 + t = 4.242640687 + -3 t = 4.242640687 + -3 Combine like terms: 4.242640687 + -3 = 1.242640687 t = 1.242640687 Simplifying t = 1.242640687Subproblem 2
t + 3 = -4.242640687 Simplifying t + 3 = -4.242640687 Reorder the terms: 3 + t = -4.242640687 Solving 3 + t = -4.242640687 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 = -4.242640687 + -3 Combine like terms: 3 + -3 = 0 0 + t = -4.242640687 + -3 t = -4.242640687 + -3 Combine like terms: -4.242640687 + -3 = -7.242640687 t = -7.242640687 Simplifying t = -7.242640687Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.242640687, -7.242640687}Solution
t = {1.242640687, -7.242640687}
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