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Simplifying 2g5(-1g5 + 6g3 + -1g) = 0 Reorder the terms: 2g5(-1g + 6g3 + -1g5) = 0 (-1g * 2g5 + 6g3 * 2g5 + -1g5 * 2g5) = 0 (-2g6 + 12g8 + -2g10) = 0 Solving -2g6 + 12g8 + -2g10 = 0 Solving for variable 'g'. Factor out the Greatest Common Factor (GCF), '2g6'. 2g6(-1 + 6g2 + -1g4) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'g6' equal to zero and attempt to solve: Simplifying g6 = 0 Solving g6 = 0 Move all terms containing g to the left, all other terms to the right. Simplifying g6 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(-1 + 6g2 + -1g4)' equal to zero and attempt to solve: Simplifying -1 + 6g2 + -1g4 = 0 Solving -1 + 6g2 + -1g4 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -6g2 + g4 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -6g2 + -1 + g4 = 0 + -1 Reorder the terms: 1 + -1 + -6g2 + g4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -6g2 + g4 = 0 + -1 -6g2 + g4 = 0 + -1 Combine like terms: 0 + -1 = -1 -6g2 + g4 = -1 The g term is -6g2. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6g2 + 9 + g4 = -1 + 9 Reorder the terms: 9 + -6g2 + g4 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + -6g2 + g4 = 8 Factor a perfect square on the left side: (g2 + -3)(g2 + -3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (g2 + -3) equal to 2.828427125 and -2.828427125.Subproblem 1
g2 + -3 = 2.828427125 Simplifying g2 + -3 = 2.828427125 Reorder the terms: -3 + g2 = 2.828427125 Solving -3 + g2 = 2.828427125 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + g2 = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + g2 = 2.828427125 + 3 g2 = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 g2 = 5.828427125 Simplifying g2 = 5.828427125 Take the square root of each side: g = {-2.414213562, 2.414213562}Subproblem 2
g2 + -3 = -2.828427125 Simplifying g2 + -3 = -2.828427125 Reorder the terms: -3 + g2 = -2.828427125 Solving -3 + g2 = -2.828427125 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + g2 = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + g2 = -2.828427125 + 3 g2 = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 g2 = 0.171572875 Simplifying g2 = 0.171572875 Take the square root of each side: g = {-0.414213562, 0.414213562}Solution
The solution to the problem is based on the solutions from the subproblems. g = {-2.414213562, 2.414213562, -0.414213562, 0.414213562}Solution
g = {-2.414213562, 2.414213562, -0.414213562, 0.414213562}
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