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Simplifying (-6v2 + -2v + -8) + -1(-2v2 + 6v + 1) + (6v2 + 4v + -5) = 0 Reorder the terms: (-8 + -2v + -6v2) + -1(-2v2 + 6v + 1) + (6v2 + 4v + -5) = 0 Remove parenthesis around (-8 + -2v + -6v2) -8 + -2v + -6v2 + -1(-2v2 + 6v + 1) + (6v2 + 4v + -5) = 0 Reorder the terms: -8 + -2v + -6v2 + -1(1 + 6v + -2v2) + (6v2 + 4v + -5) = 0 -8 + -2v + -6v2 + (1 * -1 + 6v * -1 + -2v2 * -1) + (6v2 + 4v + -5) = 0 -8 + -2v + -6v2 + (-1 + -6v + 2v2) + (6v2 + 4v + -5) = 0 Reorder the terms: -8 + -2v + -6v2 + -1 + -6v + 2v2 + (-5 + 4v + 6v2) = 0 Remove parenthesis around (-5 + 4v + 6v2) -8 + -2v + -6v2 + -1 + -6v + 2v2 + -5 + 4v + 6v2 = 0 Reorder the terms: -8 + -1 + -5 + -2v + -6v + 4v + -6v2 + 2v2 + 6v2 = 0 Combine like terms: -8 + -1 = -9 -9 + -5 + -2v + -6v + 4v + -6v2 + 2v2 + 6v2 = 0 Combine like terms: -9 + -5 = -14 -14 + -2v + -6v + 4v + -6v2 + 2v2 + 6v2 = 0 Combine like terms: -2v + -6v = -8v -14 + -8v + 4v + -6v2 + 2v2 + 6v2 = 0 Combine like terms: -8v + 4v = -4v -14 + -4v + -6v2 + 2v2 + 6v2 = 0 Combine like terms: -6v2 + 2v2 = -4v2 -14 + -4v + -4v2 + 6v2 = 0 Combine like terms: -4v2 + 6v2 = 2v2 -14 + -4v + 2v2 = 0 Solving -14 + -4v + 2v2 = 0 Solving for variable 'v'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -2v + v2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -2v + v2)' equal to zero and attempt to solve: Simplifying -7 + -2v + v2 = 0 Solving -7 + -2v + v2 = 0 Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + -2v + 7 + v2 = 0 + 7 Reorder the terms: -7 + 7 + -2v + v2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -2v + v2 = 0 + 7 -2v + v2 = 0 + 7 Combine like terms: 0 + 7 = 7 -2v + v2 = 7 The v term is -2v. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2v + 1 + v2 = 7 + 1 Reorder the terms: 1 + -2v + v2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + -2v + v2 = 8 Factor a perfect square on the left side: (v + -1)(v + -1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (v + -1) equal to 2.828427125 and -2.828427125.Subproblem 1
v + -1 = 2.828427125 Simplifying v + -1 = 2.828427125 Reorder the terms: -1 + v = 2.828427125 Solving -1 + v = 2.828427125 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + v = 2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + v = 2.828427125 + 1 v = 2.828427125 + 1 Combine like terms: 2.828427125 + 1 = 3.828427125 v = 3.828427125 Simplifying v = 3.828427125Subproblem 2
v + -1 = -2.828427125 Simplifying v + -1 = -2.828427125 Reorder the terms: -1 + v = -2.828427125 Solving -1 + v = -2.828427125 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + v = -2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + v = -2.828427125 + 1 v = -2.828427125 + 1 Combine like terms: -2.828427125 + 1 = -1.828427125 v = -1.828427125 Simplifying v = -1.828427125Solution
The solution to the problem is based on the solutions from the subproblems. v = {3.828427125, -1.828427125}Solution
v = {3.828427125, -1.828427125}
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