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Simplifying 8v2 + 5v + -4 = v2 + 7v + 7 Reorder the terms: -4 + 5v + 8v2 = v2 + 7v + 7 Reorder the terms: -4 + 5v + 8v2 = 7 + 7v + v2 Solving -4 + 5v + 8v2 = 7 + 7v + v2 Solving for variable 'v'. Reorder the terms: -4 + -7 + 5v + -7v + 8v2 + -1v2 = 7 + 7v + v2 + -7 + -7v + -1v2 Combine like terms: -4 + -7 = -11 -11 + 5v + -7v + 8v2 + -1v2 = 7 + 7v + v2 + -7 + -7v + -1v2 Combine like terms: 5v + -7v = -2v -11 + -2v + 8v2 + -1v2 = 7 + 7v + v2 + -7 + -7v + -1v2 Combine like terms: 8v2 + -1v2 = 7v2 -11 + -2v + 7v2 = 7 + 7v + v2 + -7 + -7v + -1v2 Reorder the terms: -11 + -2v + 7v2 = 7 + -7 + 7v + -7v + v2 + -1v2 Combine like terms: 7 + -7 = 0 -11 + -2v + 7v2 = 0 + 7v + -7v + v2 + -1v2 -11 + -2v + 7v2 = 7v + -7v + v2 + -1v2 Combine like terms: 7v + -7v = 0 -11 + -2v + 7v2 = 0 + v2 + -1v2 -11 + -2v + 7v2 = v2 + -1v2 Combine like terms: v2 + -1v2 = 0 -11 + -2v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.571428571 + -0.2857142857v + v2 = 0 Move the constant term to the right: Add '1.571428571' to each side of the equation. -1.571428571 + -0.2857142857v + 1.571428571 + v2 = 0 + 1.571428571 Reorder the terms: -1.571428571 + 1.571428571 + -0.2857142857v + v2 = 0 + 1.571428571 Combine like terms: -1.571428571 + 1.571428571 = 0.000000000 0.000000000 + -0.2857142857v + v2 = 0 + 1.571428571 -0.2857142857v + v2 = 0 + 1.571428571 Combine like terms: 0 + 1.571428571 = 1.571428571 -0.2857142857v + v2 = 1.571428571 The v term is -0.2857142857v. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857v + 0.02040816328 + v2 = 1.571428571 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857v + v2 = 1.571428571 + 0.02040816328 Combine like terms: 1.571428571 + 0.02040816328 = 1.59183673428 0.02040816328 + -0.2857142857v + v2 = 1.59183673428 Factor a perfect square on the left side: (v + -0.1428571429)(v + -0.1428571429) = 1.59183673428 Calculate the square root of the right side: 1.261680124 Break this problem into two subproblems by setting (v + -0.1428571429) equal to 1.261680124 and -1.261680124.Subproblem 1
v + -0.1428571429 = 1.261680124 Simplifying v + -0.1428571429 = 1.261680124 Reorder the terms: -0.1428571429 + v = 1.261680124 Solving -0.1428571429 + v = 1.261680124 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + v = 1.261680124 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + v = 1.261680124 + 0.1428571429 v = 1.261680124 + 0.1428571429 Combine like terms: 1.261680124 + 0.1428571429 = 1.4045372669 v = 1.4045372669 Simplifying v = 1.4045372669Subproblem 2
v + -0.1428571429 = -1.261680124 Simplifying v + -0.1428571429 = -1.261680124 Reorder the terms: -0.1428571429 + v = -1.261680124 Solving -0.1428571429 + v = -1.261680124 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + v = -1.261680124 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + v = -1.261680124 + 0.1428571429 v = -1.261680124 + 0.1428571429 Combine like terms: -1.261680124 + 0.1428571429 = -1.1188229811 v = -1.1188229811 Simplifying v = -1.1188229811Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.4045372669, -1.1188229811}
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