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Simplifying 6v3 + 15v2 + 5v = 0 Reorder the terms: 5v + 15v2 + 6v3 = 0 Solving 5v + 15v2 + 6v3 = 0 Solving for variable 'v'. Factor out the Greatest Common Factor (GCF), 'v'. v(5 + 15v + 6v2) = 0Subproblem 1
Set the factor 'v' equal to zero and attempt to solve: Simplifying v = 0 Solving v = 0 Move all terms containing v to the left, all other terms to the right. Simplifying v = 0Subproblem 2
Set the factor '(5 + 15v + 6v2)' equal to zero and attempt to solve: Simplifying 5 + 15v + 6v2 = 0 Solving 5 + 15v + 6v2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + 2.5v + v2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + 2.5v + -0.8333333333 + v2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + 2.5v + v2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + 2.5v + v2 = 0 + -0.8333333333 2.5v + v2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 2.5v + v2 = -0.8333333333 The v term is 2.5v. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5v + 1.5625 + v2 = -0.8333333333 + 1.5625 Reorder the terms: 1.5625 + 2.5v + v2 = -0.8333333333 + 1.5625 Combine like terms: -0.8333333333 + 1.5625 = 0.7291666667 1.5625 + 2.5v + v2 = 0.7291666667 Factor a perfect square on the left side: (v + 1.25)(v + 1.25) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (v + 1.25) equal to 0.853912564 and -0.853912564.Subproblem 1
v + 1.25 = 0.853912564 Simplifying v + 1.25 = 0.853912564 Reorder the terms: 1.25 + v = 0.853912564 Solving 1.25 + v = 0.853912564 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + v = 0.853912564 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + v = 0.853912564 + -1.25 v = 0.853912564 + -1.25 Combine like terms: 0.853912564 + -1.25 = -0.396087436 v = -0.396087436 Simplifying v = -0.396087436Subproblem 2
v + 1.25 = -0.853912564 Simplifying v + 1.25 = -0.853912564 Reorder the terms: 1.25 + v = -0.853912564 Solving 1.25 + v = -0.853912564 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + v = -0.853912564 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + v = -0.853912564 + -1.25 v = -0.853912564 + -1.25 Combine like terms: -0.853912564 + -1.25 = -2.103912564 v = -2.103912564 Simplifying v = -2.103912564Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.396087436, -2.103912564}Solution
v = {0, -0.396087436, -2.103912564}
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