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Simplifying 17v2 + 12v + 3(8v + -5) = 0 Reorder the terms: 17v2 + 12v + 3(-5 + 8v) = 0 17v2 + 12v + (-5 * 3 + 8v * 3) = 0 17v2 + 12v + (-15 + 24v) = 0 Reorder the terms: -15 + 12v + 24v + 17v2 = 0 Combine like terms: 12v + 24v = 36v -15 + 36v + 17v2 = 0 Solving -15 + 36v + 17v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 17 the coefficient of the squared term: Divide each side by '17'. -0.8823529412 + 2.117647059v + v2 = 0 Move the constant term to the right: Add '0.8823529412' to each side of the equation. -0.8823529412 + 2.117647059v + 0.8823529412 + v2 = 0 + 0.8823529412 Reorder the terms: -0.8823529412 + 0.8823529412 + 2.117647059v + v2 = 0 + 0.8823529412 Combine like terms: -0.8823529412 + 0.8823529412 = 0.0000000000 0.0000000000 + 2.117647059v + v2 = 0 + 0.8823529412 2.117647059v + v2 = 0 + 0.8823529412 Combine like terms: 0 + 0.8823529412 = 0.8823529412 2.117647059v + v2 = 0.8823529412 The v term is 2.117647059v. Take half its coefficient (1.05882353). Square it (1.121107268) and add it to both sides. Add '1.121107268' to each side of the equation. 2.117647059v + 1.121107268 + v2 = 0.8823529412 + 1.121107268 Reorder the terms: 1.121107268 + 2.117647059v + v2 = 0.8823529412 + 1.121107268 Combine like terms: 0.8823529412 + 1.121107268 = 2.0034602092 1.121107268 + 2.117647059v + v2 = 2.0034602092 Factor a perfect square on the left side: (v + 1.05882353)(v + 1.05882353) = 2.0034602092 Calculate the square root of the right side: 1.415436402 Break this problem into two subproblems by setting (v + 1.05882353) equal to 1.415436402 and -1.415436402.Subproblem 1
v + 1.05882353 = 1.415436402 Simplifying v + 1.05882353 = 1.415436402 Reorder the terms: 1.05882353 + v = 1.415436402 Solving 1.05882353 + v = 1.415436402 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.05882353' to each side of the equation. 1.05882353 + -1.05882353 + v = 1.415436402 + -1.05882353 Combine like terms: 1.05882353 + -1.05882353 = 0.00000000 0.00000000 + v = 1.415436402 + -1.05882353 v = 1.415436402 + -1.05882353 Combine like terms: 1.415436402 + -1.05882353 = 0.356612872 v = 0.356612872 Simplifying v = 0.356612872Subproblem 2
v + 1.05882353 = -1.415436402 Simplifying v + 1.05882353 = -1.415436402 Reorder the terms: 1.05882353 + v = -1.415436402 Solving 1.05882353 + v = -1.415436402 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.05882353' to each side of the equation. 1.05882353 + -1.05882353 + v = -1.415436402 + -1.05882353 Combine like terms: 1.05882353 + -1.05882353 = 0.00000000 0.00000000 + v = -1.415436402 + -1.05882353 v = -1.415436402 + -1.05882353 Combine like terms: -1.415436402 + -1.05882353 = -2.474259932 v = -2.474259932 Simplifying v = -2.474259932Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.356612872, -2.474259932}
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