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Simplifying 12t + 9 = 9t2 Reorder the terms: 9 + 12t = 9t2 Solving 9 + 12t = 9t2 Solving for variable 't'. Combine like terms: 9t2 + -9t2 = 0 9 + 12t + -9t2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(3 + 4t + -3t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(3 + 4t + -3t2)' equal to zero and attempt to solve: Simplifying 3 + 4t + -3t2 = 0 Solving 3 + 4t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1 + -1.333333333t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.333333333t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -1.333333333t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.333333333t + t2 = 0 + 1 -1.333333333t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1.333333333t + t2 = 1 The t term is -1.333333333t. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333t + 0.4444444442 + t2 = 1 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333t + t2 = 1 + 0.4444444442 Combine like terms: 1 + 0.4444444442 = 1.4444444442 0.4444444442 + -1.333333333t + t2 = 1.4444444442 Factor a perfect square on the left side: (t + -0.6666666665)(t + -0.6666666665) = 1.4444444442 Calculate the square root of the right side: 1.201850425 Break this problem into two subproblems by setting (t + -0.6666666665) equal to 1.201850425 and -1.201850425.Subproblem 1
t + -0.6666666665 = 1.201850425 Simplifying t + -0.6666666665 = 1.201850425 Reorder the terms: -0.6666666665 + t = 1.201850425 Solving -0.6666666665 + t = 1.201850425 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + t = 1.201850425 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + t = 1.201850425 + 0.6666666665 t = 1.201850425 + 0.6666666665 Combine like terms: 1.201850425 + 0.6666666665 = 1.8685170915 t = 1.8685170915 Simplifying t = 1.8685170915Subproblem 2
t + -0.6666666665 = -1.201850425 Simplifying t + -0.6666666665 = -1.201850425 Reorder the terms: -0.6666666665 + t = -1.201850425 Solving -0.6666666665 + t = -1.201850425 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + t = -1.201850425 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + t = -1.201850425 + 0.6666666665 t = -1.201850425 + 0.6666666665 Combine like terms: -1.201850425 + 0.6666666665 = -0.5351837585 t = -0.5351837585 Simplifying t = -0.5351837585Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.8685170915, -0.5351837585}Solution
t = {1.8685170915, -0.5351837585}
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