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Simplifying 2x2 + 10x + 11 = 0 Reorder the terms: 11 + 10x + 2x2 = 0 Solving 11 + 10x + 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'. 5.5 + 5x + x2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + 5x + -5.5 + x2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + 5x + x2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + 5x + x2 = 0 + -5.5 5x + x2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 5x + x2 = -5.5 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = -5.5 + 6.25 Reorder the terms: 6.25 + 5x + x2 = -5.5 + 6.25 Combine like terms: -5.5 + 6.25 = 0.75 6.25 + 5x + x2 = 0.75 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 2.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 2.5 = 0.866025404 Simplifying x + 2.5 = 0.866025404 Reorder the terms: 2.5 + x = 0.866025404 Solving 2.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 0.866025404 + -2.5 x = 0.866025404 + -2.5 Combine like terms: 0.866025404 + -2.5 = -1.633974596 x = -1.633974596 Simplifying x = -1.633974596Subproblem 2
x + 2.5 = -0.866025404 Simplifying x + 2.5 = -0.866025404 Reorder the terms: 2.5 + x = -0.866025404 Solving 2.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -0.866025404 + -2.5 x = -0.866025404 + -2.5 Combine like terms: -0.866025404 + -2.5 = -3.366025404 x = -3.366025404 Simplifying x = -3.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.633974596, -3.366025404}
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