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Simplifying 4v2 + 12v + -7v = 24 Reorder the terms: 12v + -7v + 4v2 = 24 Combine like terms: 12v + -7v = 5v 5v + 4v2 = 24 Solving 5v + 4v2 = 24 Solving for variable 'v'. Reorder the terms: -24 + 5v + 4v2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 5v + 4v2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6 + 1.25v + v2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 1.25v + 6 + v2 = 0 + 6 Reorder the terms: -6 + 6 + 1.25v + v2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 1.25v + v2 = 0 + 6 1.25v + v2 = 0 + 6 Combine like terms: 0 + 6 = 6 1.25v + v2 = 6 The v term is 1.25v. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25v + 0.390625 + v2 = 6 + 0.390625 Reorder the terms: 0.390625 + 1.25v + v2 = 6 + 0.390625 Combine like terms: 6 + 0.390625 = 6.390625 0.390625 + 1.25v + v2 = 6.390625 Factor a perfect square on the left side: (v + 0.625)(v + 0.625) = 6.390625 Calculate the square root of the right side: 2.527968552 Break this problem into two subproblems by setting (v + 0.625) equal to 2.527968552 and -2.527968552.Subproblem 1
v + 0.625 = 2.527968552 Simplifying v + 0.625 = 2.527968552 Reorder the terms: 0.625 + v = 2.527968552 Solving 0.625 + v = 2.527968552 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + v = 2.527968552 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + v = 2.527968552 + -0.625 v = 2.527968552 + -0.625 Combine like terms: 2.527968552 + -0.625 = 1.902968552 v = 1.902968552 Simplifying v = 1.902968552Subproblem 2
v + 0.625 = -2.527968552 Simplifying v + 0.625 = -2.527968552 Reorder the terms: 0.625 + v = -2.527968552 Solving 0.625 + v = -2.527968552 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + v = -2.527968552 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + v = -2.527968552 + -0.625 v = -2.527968552 + -0.625 Combine like terms: -2.527968552 + -0.625 = -3.152968552 v = -3.152968552 Simplifying v = -3.152968552Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.902968552, -3.152968552}
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