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Simplifying 7v2 + -42 = -32v Reorder the terms: -42 + 7v2 = -32v Solving -42 + 7v2 = -32v Solving for variable 'v'. Reorder the terms: -42 + 32v + 7v2 = -32v + 32v Combine like terms: -32v + 32v = 0 -42 + 32v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -6 + 4.571428571v + v2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 4.571428571v + 6 + v2 = 0 + 6 Reorder the terms: -6 + 6 + 4.571428571v + v2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 4.571428571v + v2 = 0 + 6 4.571428571v + v2 = 0 + 6 Combine like terms: 0 + 6 = 6 4.571428571v + v2 = 6 The v term is 4.571428571v. Take half its coefficient (2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. 4.571428571v + 5.224489797 + v2 = 6 + 5.224489797 Reorder the terms: 5.224489797 + 4.571428571v + v2 = 6 + 5.224489797 Combine like terms: 6 + 5.224489797 = 11.224489797 5.224489797 + 4.571428571v + v2 = 11.224489797 Factor a perfect square on the left side: (v + 2.285714286)(v + 2.285714286) = 11.224489797 Calculate the square root of the right side: 3.350296971 Break this problem into two subproblems by setting (v + 2.285714286) equal to 3.350296971 and -3.350296971.Subproblem 1
v + 2.285714286 = 3.350296971 Simplifying v + 2.285714286 = 3.350296971 Reorder the terms: 2.285714286 + v = 3.350296971 Solving 2.285714286 + v = 3.350296971 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + v = 3.350296971 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + v = 3.350296971 + -2.285714286 v = 3.350296971 + -2.285714286 Combine like terms: 3.350296971 + -2.285714286 = 1.064582685 v = 1.064582685 Simplifying v = 1.064582685Subproblem 2
v + 2.285714286 = -3.350296971 Simplifying v + 2.285714286 = -3.350296971 Reorder the terms: 2.285714286 + v = -3.350296971 Solving 2.285714286 + v = -3.350296971 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + v = -3.350296971 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + v = -3.350296971 + -2.285714286 v = -3.350296971 + -2.285714286 Combine like terms: -3.350296971 + -2.285714286 = -5.636011257 v = -5.636011257 Simplifying v = -5.636011257Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.064582685, -5.636011257}
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