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Simplifying (v + 7)(v + 7) = 2v2 + 16v + 3v Reorder the terms: (7 + v)(v + 7) = 2v2 + 16v + 3v Reorder the terms: (7 + v)(7 + v) = 2v2 + 16v + 3v Multiply (7 + v) * (7 + v) (7(7 + v) + v(7 + v)) = 2v2 + 16v + 3v ((7 * 7 + v * 7) + v(7 + v)) = 2v2 + 16v + 3v ((49 + 7v) + v(7 + v)) = 2v2 + 16v + 3v (49 + 7v + (7 * v + v * v)) = 2v2 + 16v + 3v (49 + 7v + (7v + v2)) = 2v2 + 16v + 3v Combine like terms: 7v + 7v = 14v (49 + 14v + v2) = 2v2 + 16v + 3v Reorder the terms: 49 + 14v + v2 = 16v + 3v + 2v2 Combine like terms: 16v + 3v = 19v 49 + 14v + v2 = 19v + 2v2 Solving 49 + 14v + v2 = 19v + 2v2 Solving for variable 'v'. Reorder the terms: 49 + 14v + -19v + v2 + -2v2 = 19v + 2v2 + -19v + -2v2 Combine like terms: 14v + -19v = -5v 49 + -5v + v2 + -2v2 = 19v + 2v2 + -19v + -2v2 Combine like terms: v2 + -2v2 = -1v2 49 + -5v + -1v2 = 19v + 2v2 + -19v + -2v2 Reorder the terms: 49 + -5v + -1v2 = 19v + -19v + 2v2 + -2v2 Combine like terms: 19v + -19v = 0 49 + -5v + -1v2 = 0 + 2v2 + -2v2 49 + -5v + -1v2 = 2v2 + -2v2 Combine like terms: 2v2 + -2v2 = 0 49 + -5v + -1v2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -49 + 5v + v2 = 0 Move the constant term to the right: Add '49' to each side of the equation. -49 + 5v + 49 + v2 = 0 + 49 Reorder the terms: -49 + 49 + 5v + v2 = 0 + 49 Combine like terms: -49 + 49 = 0 0 + 5v + v2 = 0 + 49 5v + v2 = 0 + 49 Combine like terms: 0 + 49 = 49 5v + v2 = 49 The v term is 5v. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5v + 6.25 + v2 = 49 + 6.25 Reorder the terms: 6.25 + 5v + v2 = 49 + 6.25 Combine like terms: 49 + 6.25 = 55.25 6.25 + 5v + v2 = 55.25 Factor a perfect square on the left side: (v + 2.5)(v + 2.5) = 55.25 Calculate the square root of the right side: 7.433034374 Break this problem into two subproblems by setting (v + 2.5) equal to 7.433034374 and -7.433034374.Subproblem 1
v + 2.5 = 7.433034374 Simplifying v + 2.5 = 7.433034374 Reorder the terms: 2.5 + v = 7.433034374 Solving 2.5 + v = 7.433034374 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + v = 7.433034374 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + v = 7.433034374 + -2.5 v = 7.433034374 + -2.5 Combine like terms: 7.433034374 + -2.5 = 4.933034374 v = 4.933034374 Simplifying v = 4.933034374Subproblem 2
v + 2.5 = -7.433034374 Simplifying v + 2.5 = -7.433034374 Reorder the terms: 2.5 + v = -7.433034374 Solving 2.5 + v = -7.433034374 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + v = -7.433034374 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + v = -7.433034374 + -2.5 v = -7.433034374 + -2.5 Combine like terms: -7.433034374 + -2.5 = -9.933034374 v = -9.933034374 Simplifying v = -9.933034374Solution
The solution to the problem is based on the solutions from the subproblems. v = {4.933034374, -9.933034374}
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