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Simplifying 4 + -3v(2 + 5v) + 7 = 0 4 + (2 * -3v + 5v * -3v) + 7 = 0 4 + (-6v + -15v2) + 7 = 0 Reorder the terms: 4 + 7 + -6v + -15v2 = 0 Combine like terms: 4 + 7 = 11 11 + -6v + -15v2 = 0 Solving 11 + -6v + -15v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -0.7333333333 + 0.4v + v2 = 0 Move the constant term to the right: Add '0.7333333333' to each side of the equation. -0.7333333333 + 0.4v + 0.7333333333 + v2 = 0 + 0.7333333333 Reorder the terms: -0.7333333333 + 0.7333333333 + 0.4v + v2 = 0 + 0.7333333333 Combine like terms: -0.7333333333 + 0.7333333333 = 0.0000000000 0.0000000000 + 0.4v + v2 = 0 + 0.7333333333 0.4v + v2 = 0 + 0.7333333333 Combine like terms: 0 + 0.7333333333 = 0.7333333333 0.4v + v2 = 0.7333333333 The v term is 0.4v. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4v + 0.04 + v2 = 0.7333333333 + 0.04 Reorder the terms: 0.04 + 0.4v + v2 = 0.7333333333 + 0.04 Combine like terms: 0.7333333333 + 0.04 = 0.7733333333 0.04 + 0.4v + v2 = 0.7733333333 Factor a perfect square on the left side: (v + 0.2)(v + 0.2) = 0.7733333333 Calculate the square root of the right side: 0.879393731 Break this problem into two subproblems by setting (v + 0.2) equal to 0.879393731 and -0.879393731.Subproblem 1
v + 0.2 = 0.879393731 Simplifying v + 0.2 = 0.879393731 Reorder the terms: 0.2 + v = 0.879393731 Solving 0.2 + v = 0.879393731 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + v = 0.879393731 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + v = 0.879393731 + -0.2 v = 0.879393731 + -0.2 Combine like terms: 0.879393731 + -0.2 = 0.679393731 v = 0.679393731 Simplifying v = 0.679393731Subproblem 2
v + 0.2 = -0.879393731 Simplifying v + 0.2 = -0.879393731 Reorder the terms: 0.2 + v = -0.879393731 Solving 0.2 + v = -0.879393731 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + v = -0.879393731 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + v = -0.879393731 + -0.2 v = -0.879393731 + -0.2 Combine like terms: -0.879393731 + -0.2 = -1.079393731 v = -1.079393731 Simplifying v = -1.079393731Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.679393731, -1.079393731}
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