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Simplifying 7v2 + -53v + -27 = 3 Reorder the terms: -27 + -53v + 7v2 = 3 Solving -27 + -53v + 7v2 = 3 Solving for variable 'v'. Reorder the terms: -27 + -3 + -53v + 7v2 = 3 + -3 Combine like terms: -27 + -3 = -30 -30 + -53v + 7v2 = 3 + -3 Combine like terms: 3 + -3 = 0 -30 + -53v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.285714286 + -7.571428571v + v2 = 0 Move the constant term to the right: Add '4.285714286' to each side of the equation. -4.285714286 + -7.571428571v + 4.285714286 + v2 = 0 + 4.285714286 Reorder the terms: -4.285714286 + 4.285714286 + -7.571428571v + v2 = 0 + 4.285714286 Combine like terms: -4.285714286 + 4.285714286 = 0.000000000 0.000000000 + -7.571428571v + v2 = 0 + 4.285714286 -7.571428571v + v2 = 0 + 4.285714286 Combine like terms: 0 + 4.285714286 = 4.285714286 -7.571428571v + v2 = 4.285714286 The v term is -7.571428571v. Take half its coefficient (-3.785714286). Square it (14.33163266) and add it to both sides. Add '14.33163266' to each side of the equation. -7.571428571v + 14.33163266 + v2 = 4.285714286 + 14.33163266 Reorder the terms: 14.33163266 + -7.571428571v + v2 = 4.285714286 + 14.33163266 Combine like terms: 4.285714286 + 14.33163266 = 18.617346946 14.33163266 + -7.571428571v + v2 = 18.617346946 Factor a perfect square on the left side: (v + -3.785714286)(v + -3.785714286) = 18.617346946 Calculate the square root of the right side: 4.314782375 Break this problem into two subproblems by setting (v + -3.785714286) equal to 4.314782375 and -4.314782375.Subproblem 1
v + -3.785714286 = 4.314782375 Simplifying v + -3.785714286 = 4.314782375 Reorder the terms: -3.785714286 + v = 4.314782375 Solving -3.785714286 + v = 4.314782375 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.785714286' to each side of the equation. -3.785714286 + 3.785714286 + v = 4.314782375 + 3.785714286 Combine like terms: -3.785714286 + 3.785714286 = 0.000000000 0.000000000 + v = 4.314782375 + 3.785714286 v = 4.314782375 + 3.785714286 Combine like terms: 4.314782375 + 3.785714286 = 8.100496661 v = 8.100496661 Simplifying v = 8.100496661Subproblem 2
v + -3.785714286 = -4.314782375 Simplifying v + -3.785714286 = -4.314782375 Reorder the terms: -3.785714286 + v = -4.314782375 Solving -3.785714286 + v = -4.314782375 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.785714286' to each side of the equation. -3.785714286 + 3.785714286 + v = -4.314782375 + 3.785714286 Combine like terms: -3.785714286 + 3.785714286 = 0.000000000 0.000000000 + v = -4.314782375 + 3.785714286 v = -4.314782375 + 3.785714286 Combine like terms: -4.314782375 + 3.785714286 = -0.529068089 v = -0.529068089 Simplifying v = -0.529068089Solution
The solution to the problem is based on the solutions from the subproblems. v = {8.100496661, -0.529068089}
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