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Simplifying (-6v2 + 5v + 2) + (-9v2 + -7v + 3) + -1(-6v2 + 9v + 1) = 0 Reorder the terms: (2 + 5v + -6v2) + (-9v2 + -7v + 3) + -1(-6v2 + 9v + 1) = 0 Remove parenthesis around (2 + 5v + -6v2) 2 + 5v + -6v2 + (-9v2 + -7v + 3) + -1(-6v2 + 9v + 1) = 0 Reorder the terms: 2 + 5v + -6v2 + (3 + -7v + -9v2) + -1(-6v2 + 9v + 1) = 0 Remove parenthesis around (3 + -7v + -9v2) 2 + 5v + -6v2 + 3 + -7v + -9v2 + -1(-6v2 + 9v + 1) = 0 Reorder the terms: 2 + 5v + -6v2 + 3 + -7v + -9v2 + -1(1 + 9v + -6v2) = 0 2 + 5v + -6v2 + 3 + -7v + -9v2 + (1 * -1 + 9v * -1 + -6v2 * -1) = 0 2 + 5v + -6v2 + 3 + -7v + -9v2 + (-1 + -9v + 6v2) = 0 Reorder the terms: 2 + 3 + -1 + 5v + -7v + -9v + -6v2 + -9v2 + 6v2 = 0 Combine like terms: 2 + 3 = 5 5 + -1 + 5v + -7v + -9v + -6v2 + -9v2 + 6v2 = 0 Combine like terms: 5 + -1 = 4 4 + 5v + -7v + -9v + -6v2 + -9v2 + 6v2 = 0 Combine like terms: 5v + -7v = -2v 4 + -2v + -9v + -6v2 + -9v2 + 6v2 = 0 Combine like terms: -2v + -9v = -11v 4 + -11v + -6v2 + -9v2 + 6v2 = 0 Combine like terms: -6v2 + -9v2 = -15v2 4 + -11v + -15v2 + 6v2 = 0 Combine like terms: -15v2 + 6v2 = -9v2 4 + -11v + -9v2 = 0 Solving 4 + -11v + -9v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.4444444444 + 1.222222222v + v2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 1.222222222v + 0.4444444444 + v2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 1.222222222v + v2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 1.222222222v + v2 = 0 + 0.4444444444 1.222222222v + v2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 1.222222222v + v2 = 0.4444444444 The v term is 1.222222222v. Take half its coefficient (0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. 1.222222222v + 0.3734567900 + v2 = 0.4444444444 + 0.3734567900 Reorder the terms: 0.3734567900 + 1.222222222v + v2 = 0.4444444444 + 0.3734567900 Combine like terms: 0.4444444444 + 0.3734567900 = 0.8179012344 0.3734567900 + 1.222222222v + v2 = 0.8179012344 Factor a perfect square on the left side: (v + 0.611111111)(v + 0.611111111) = 0.8179012344 Calculate the square root of the right side: 0.904378922 Break this problem into two subproblems by setting (v + 0.611111111) equal to 0.904378922 and -0.904378922.Subproblem 1
v + 0.611111111 = 0.904378922 Simplifying v + 0.611111111 = 0.904378922 Reorder the terms: 0.611111111 + v = 0.904378922 Solving 0.611111111 + v = 0.904378922 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + v = 0.904378922 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + v = 0.904378922 + -0.611111111 v = 0.904378922 + -0.611111111 Combine like terms: 0.904378922 + -0.611111111 = 0.293267811 v = 0.293267811 Simplifying v = 0.293267811Subproblem 2
v + 0.611111111 = -0.904378922 Simplifying v + 0.611111111 = -0.904378922 Reorder the terms: 0.611111111 + v = -0.904378922 Solving 0.611111111 + v = -0.904378922 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + v = -0.904378922 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + v = -0.904378922 + -0.611111111 v = -0.904378922 + -0.611111111 Combine like terms: -0.904378922 + -0.611111111 = -1.515490033 v = -1.515490033 Simplifying v = -1.515490033Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.293267811, -1.515490033}
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