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Simplifying 24 = 3t2 + 18t Reorder the terms: 24 = 18t + 3t2 Solving 24 = 18t + 3t2 Solving for variable 't'. Reorder the terms: 24 + -18t + -3t2 = 18t + -18t + 3t2 + -3t2 Combine like terms: 18t + -18t = 0 24 + -18t + -3t2 = 0 + 3t2 + -3t2 24 + -18t + -3t2 = 3t2 + -3t2 Combine like terms: 3t2 + -3t2 = 0 24 + -18t + -3t2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(8 + -6t + -1t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(8 + -6t + -1t2)' equal to zero and attempt to solve: Simplifying 8 + -6t + -1t2 = 0 Solving 8 + -6t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -8 + 6t + t2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 6t + 8 + t2 = 0 + 8 Reorder the terms: -8 + 8 + 6t + t2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 6t + t2 = 0 + 8 6t + t2 = 0 + 8 Combine like terms: 0 + 8 = 8 6t + t2 = 8 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 = 8 + 9 Reorder the terms: 9 + 6t + t2 = 8 + 9 Combine like terms: 8 + 9 = 17 9 + 6t + t2 = 17 Factor a perfect square on the left side: (t + 3)(t + 3) = 17 Calculate the square root of the right side: 4.123105626 Break this problem into two subproblems by setting (t + 3) equal to 4.123105626 and -4.123105626.Subproblem 1
t + 3 = 4.123105626 Simplifying t + 3 = 4.123105626 Reorder the terms: 3 + t = 4.123105626 Solving 3 + t = 4.123105626 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 = 4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + t = 4.123105626 + -3 t = 4.123105626 + -3 Combine like terms: 4.123105626 + -3 = 1.123105626 t = 1.123105626 Simplifying t = 1.123105626Subproblem 2
t + 3 = -4.123105626 Simplifying t + 3 = -4.123105626 Reorder the terms: 3 + t = -4.123105626 Solving 3 + t = -4.123105626 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 = -4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + t = -4.123105626 + -3 t = -4.123105626 + -3 Combine like terms: -4.123105626 + -3 = -7.123105626 t = -7.123105626 Simplifying t = -7.123105626Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.123105626, -7.123105626}Solution
t = {1.123105626, -7.123105626}
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