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Simplifying 2x2 + 14x + 23 = 0 Reorder the terms: 23 + 14x + 2x2 = 0 Solving 23 + 14x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 11.5 + 7x + x2 = 0 Move the constant term to the right: Add '-11.5' to each side of the equation. 11.5 + 7x + -11.5 + x2 = 0 + -11.5 Reorder the terms: 11.5 + -11.5 + 7x + x2 = 0 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + 7x + x2 = 0 + -11.5 7x + x2 = 0 + -11.5 Combine like terms: 0 + -11.5 = -11.5 7x + x2 = -11.5 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = -11.5 + 12.25 Reorder the terms: 12.25 + 7x + x2 = -11.5 + 12.25 Combine like terms: -11.5 + 12.25 = 0.75 12.25 + 7x + x2 = 0.75 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 3.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 3.5 = 0.866025404 Simplifying x + 3.5 = 0.866025404 Reorder the terms: 3.5 + x = 0.866025404 Solving 3.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 0.866025404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 0.866025404 + -3.5 x = 0.866025404 + -3.5 Combine like terms: 0.866025404 + -3.5 = -2.633974596 x = -2.633974596 Simplifying x = -2.633974596Subproblem 2
x + 3.5 = -0.866025404 Simplifying x + 3.5 = -0.866025404 Reorder the terms: 3.5 + x = -0.866025404 Solving 3.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -0.866025404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -0.866025404 + -3.5 x = -0.866025404 + -3.5 Combine like terms: -0.866025404 + -3.5 = -4.366025404 x = -4.366025404 Simplifying x = -4.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.633974596, -4.366025404}
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