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Simplifying 16v2 + 24v + 9 = 360 Reorder the terms: 9 + 24v + 16v2 = 360 Solving 9 + 24v + 16v2 = 360 Solving for variable 'v'. Reorder the terms: 9 + -360 + 24v + 16v2 = 360 + -360 Combine like terms: 9 + -360 = -351 -351 + 24v + 16v2 = 360 + -360 Combine like terms: 360 + -360 = 0 -351 + 24v + 16v2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -21.9375 + 1.5v + v2 = 0 Move the constant term to the right: Add '21.9375' to each side of the equation. -21.9375 + 1.5v + 21.9375 + v2 = 0 + 21.9375 Reorder the terms: -21.9375 + 21.9375 + 1.5v + v2 = 0 + 21.9375 Combine like terms: -21.9375 + 21.9375 = 0.0000 0.0000 + 1.5v + v2 = 0 + 21.9375 1.5v + v2 = 0 + 21.9375 Combine like terms: 0 + 21.9375 = 21.9375 1.5v + v2 = 21.9375 The v term is 1.5v. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5v + 0.5625 + v2 = 21.9375 + 0.5625 Reorder the terms: 0.5625 + 1.5v + v2 = 21.9375 + 0.5625 Combine like terms: 21.9375 + 0.5625 = 22.5 0.5625 + 1.5v + v2 = 22.5 Factor a perfect square on the left side: (v + 0.75)(v + 0.75) = 22.5 Calculate the square root of the right side: 4.74341649 Break this problem into two subproblems by setting (v + 0.75) equal to 4.74341649 and -4.74341649.Subproblem 1
v + 0.75 = 4.74341649 Simplifying v + 0.75 = 4.74341649 Reorder the terms: 0.75 + v = 4.74341649 Solving 0.75 + v = 4.74341649 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + v = 4.74341649 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + v = 4.74341649 + -0.75 v = 4.74341649 + -0.75 Combine like terms: 4.74341649 + -0.75 = 3.99341649 v = 3.99341649 Simplifying v = 3.99341649Subproblem 2
v + 0.75 = -4.74341649 Simplifying v + 0.75 = -4.74341649 Reorder the terms: 0.75 + v = -4.74341649 Solving 0.75 + v = -4.74341649 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + v = -4.74341649 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + v = -4.74341649 + -0.75 v = -4.74341649 + -0.75 Combine like terms: -4.74341649 + -0.75 = -5.49341649 v = -5.49341649 Simplifying v = -5.49341649Solution
The solution to the problem is based on the solutions from the subproblems. v = {3.99341649, -5.49341649}
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