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Simplifying 7v2 + 14v + -27 = 0 Reorder the terms: -27 + 14v + 7v2 = 0 Solving -27 + 14v + 7v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -3.857142857 + 2v + v2 = 0 Move the constant term to the right: Add '3.857142857' to each side of the equation. -3.857142857 + 2v + 3.857142857 + v2 = 0 + 3.857142857 Reorder the terms: -3.857142857 + 3.857142857 + 2v + v2 = 0 + 3.857142857 Combine like terms: -3.857142857 + 3.857142857 = 0.000000000 0.000000000 + 2v + v2 = 0 + 3.857142857 2v + v2 = 0 + 3.857142857 Combine like terms: 0 + 3.857142857 = 3.857142857 2v + v2 = 3.857142857 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 = 3.857142857 + 1 Reorder the terms: 1 + 2v + v2 = 3.857142857 + 1 Combine like terms: 3.857142857 + 1 = 4.857142857 1 + 2v + v2 = 4.857142857 Factor a perfect square on the left side: (v + 1)(v + 1) = 4.857142857 Calculate the square root of the right side: 2.20389266 Break this problem into two subproblems by setting (v + 1) equal to 2.20389266 and -2.20389266.Subproblem 1
v + 1 = 2.20389266 Simplifying v + 1 = 2.20389266 Reorder the terms: 1 + v = 2.20389266 Solving 1 + v = 2.20389266 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 = 2.20389266 + -1 Combine like terms: 1 + -1 = 0 0 + v = 2.20389266 + -1 v = 2.20389266 + -1 Combine like terms: 2.20389266 + -1 = 1.20389266 v = 1.20389266 Simplifying v = 1.20389266Subproblem 2
v + 1 = -2.20389266 Simplifying v + 1 = -2.20389266 Reorder the terms: 1 + v = -2.20389266 Solving 1 + v = -2.20389266 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 = -2.20389266 + -1 Combine like terms: 1 + -1 = 0 0 + v = -2.20389266 + -1 v = -2.20389266 + -1 Combine like terms: -2.20389266 + -1 = -3.20389266 v = -3.20389266 Simplifying v = -3.20389266Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.20389266, -3.20389266}
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