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Simplifying 0 = 7v3 + -10v2 + 9v4 Reorder the terms: 0 = -10v2 + 7v3 + 9v4 Solving 0 = -10v2 + 7v3 + 9v4 Solving for variable 'v'. Remove the zero: 10v2 + -7v3 + -9v4 = -10v2 + 7v3 + 9v4 + 10v2 + -7v3 + -9v4 Reorder the terms: 10v2 + -7v3 + -9v4 = -10v2 + 10v2 + 7v3 + -7v3 + 9v4 + -9v4 Combine like terms: -10v2 + 10v2 = 0 10v2 + -7v3 + -9v4 = 0 + 7v3 + -7v3 + 9v4 + -9v4 10v2 + -7v3 + -9v4 = 7v3 + -7v3 + 9v4 + -9v4 Combine like terms: 7v3 + -7v3 = 0 10v2 + -7v3 + -9v4 = 0 + 9v4 + -9v4 10v2 + -7v3 + -9v4 = 9v4 + -9v4 Combine like terms: 9v4 + -9v4 = 0 10v2 + -7v3 + -9v4 = 0 Factor out the Greatest Common Factor (GCF), 'v2'. v2(10 + -7v + -9v2) = 0Subproblem 1
Set the factor 'v2' equal to zero and attempt to solve: Simplifying v2 = 0 Solving v2 = 0 Move all terms containing v to the left, all other terms to the right. Simplifying v2 = 0 Take the square root of each side: v = {0}Subproblem 2
Set the factor '(10 + -7v + -9v2)' equal to zero and attempt to solve: Simplifying 10 + -7v + -9v2 = 0 Solving 10 + -7v + -9v2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -1.111111111 + 0.7777777778v + v2 = 0 Move the constant term to the right: Add '1.111111111' to each side of the equation. -1.111111111 + 0.7777777778v + 1.111111111 + v2 = 0 + 1.111111111 Reorder the terms: -1.111111111 + 1.111111111 + 0.7777777778v + v2 = 0 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + 0.7777777778v + v2 = 0 + 1.111111111 0.7777777778v + v2 = 0 + 1.111111111 Combine like terms: 0 + 1.111111111 = 1.111111111 0.7777777778v + v2 = 1.111111111 The v term is 0.7777777778v. Take half its coefficient (0.3888888889). Square it (0.1512345679) and add it to both sides. Add '0.1512345679' to each side of the equation. 0.7777777778v + 0.1512345679 + v2 = 1.111111111 + 0.1512345679 Reorder the terms: 0.1512345679 + 0.7777777778v + v2 = 1.111111111 + 0.1512345679 Combine like terms: 1.111111111 + 0.1512345679 = 1.2623456789 0.1512345679 + 0.7777777778v + v2 = 1.2623456789 Factor a perfect square on the left side: (v + 0.3888888889)(v + 0.3888888889) = 1.2623456789 Calculate the square root of the right side: 1.123541579 Break this problem into two subproblems by setting (v + 0.3888888889) equal to 1.123541579 and -1.123541579.Subproblem 1
v + 0.3888888889 = 1.123541579 Simplifying v + 0.3888888889 = 1.123541579 Reorder the terms: 0.3888888889 + v = 1.123541579 Solving 0.3888888889 + v = 1.123541579 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + v = 1.123541579 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + v = 1.123541579 + -0.3888888889 v = 1.123541579 + -0.3888888889 Combine like terms: 1.123541579 + -0.3888888889 = 0.7346526901 v = 0.7346526901 Simplifying v = 0.7346526901Subproblem 2
v + 0.3888888889 = -1.123541579 Simplifying v + 0.3888888889 = -1.123541579 Reorder the terms: 0.3888888889 + v = -1.123541579 Solving 0.3888888889 + v = -1.123541579 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + v = -1.123541579 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + v = -1.123541579 + -0.3888888889 v = -1.123541579 + -0.3888888889 Combine like terms: -1.123541579 + -0.3888888889 = -1.5124304679 v = -1.5124304679 Simplifying v = -1.5124304679Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.7346526901, -1.5124304679}Solution
v = {0, 0.7346526901, -1.5124304679}
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