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Simplifying v2 + 15v + 54 = 3 Reorder the terms: 54 + 15v + v2 = 3 Solving 54 + 15v + v2 = 3 Solving for variable 'v'. Reorder the terms: 54 + -3 + 15v + v2 = 3 + -3 Combine like terms: 54 + -3 = 51 51 + 15v + v2 = 3 + -3 Combine like terms: 3 + -3 = 0 51 + 15v + v2 = 0 Begin completing the square. Move the constant term to the right: Add '-51' to each side of the equation. 51 + 15v + -51 + v2 = 0 + -51 Reorder the terms: 51 + -51 + 15v + v2 = 0 + -51 Combine like terms: 51 + -51 = 0 0 + 15v + v2 = 0 + -51 15v + v2 = 0 + -51 Combine like terms: 0 + -51 = -51 15v + v2 = -51 The v term is 15v. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15v + 56.25 + v2 = -51 + 56.25 Reorder the terms: 56.25 + 15v + v2 = -51 + 56.25 Combine like terms: -51 + 56.25 = 5.25 56.25 + 15v + v2 = 5.25 Factor a perfect square on the left side: (v + 7.5)(v + 7.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (v + 7.5) equal to 2.291287847 and -2.291287847.Subproblem 1
v + 7.5 = 2.291287847 Simplifying v + 7.5 = 2.291287847 Reorder the terms: 7.5 + v = 2.291287847 Solving 7.5 + v = 2.291287847 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + v = 2.291287847 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + v = 2.291287847 + -7.5 v = 2.291287847 + -7.5 Combine like terms: 2.291287847 + -7.5 = -5.208712153 v = -5.208712153 Simplifying v = -5.208712153Subproblem 2
v + 7.5 = -2.291287847 Simplifying v + 7.5 = -2.291287847 Reorder the terms: 7.5 + v = -2.291287847 Solving 7.5 + v = -2.291287847 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + v = -2.291287847 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + v = -2.291287847 + -7.5 v = -2.291287847 + -7.5 Combine like terms: -2.291287847 + -7.5 = -9.791287847 v = -9.791287847 Simplifying v = -9.791287847Solution
The solution to the problem is based on the solutions from the subproblems. v = {-5.208712153, -9.791287847}
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