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Simplifying 4x2 + 32x + 61 = 0 Reorder the terms: 61 + 32x + 4x2 = 0 Solving 61 + 32x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 15.25 + 8x + x2 = 0 Move the constant term to the right: Add '-15.25' to each side of the equation. 15.25 + 8x + -15.25 + x2 = 0 + -15.25 Reorder the terms: 15.25 + -15.25 + 8x + x2 = 0 + -15.25 Combine like terms: 15.25 + -15.25 = 0.00 0.00 + 8x + x2 = 0 + -15.25 8x + x2 = 0 + -15.25 Combine like terms: 0 + -15.25 = -15.25 8x + x2 = -15.25 The x term is 8x. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8x + 16 + x2 = -15.25 + 16 Reorder the terms: 16 + 8x + x2 = -15.25 + 16 Combine like terms: -15.25 + 16 = 0.75 16 + 8x + x2 = 0.75 Factor a perfect square on the left side: (x + 4)(x + 4) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 4) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 4 = 0.866025404 Simplifying x + 4 = 0.866025404 Reorder the terms: 4 + x = 0.866025404 Solving 4 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 0.866025404 + -4 Combine like terms: 4 + -4 = 0 0 + x = 0.866025404 + -4 x = 0.866025404 + -4 Combine like terms: 0.866025404 + -4 = -3.133974596 x = -3.133974596 Simplifying x = -3.133974596Subproblem 2
x + 4 = -0.866025404 Simplifying x + 4 = -0.866025404 Reorder the terms: 4 + x = -0.866025404 Solving 4 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -0.866025404 + -4 Combine like terms: 4 + -4 = 0 0 + x = -0.866025404 + -4 x = -0.866025404 + -4 Combine like terms: -0.866025404 + -4 = -4.866025404 x = -4.866025404 Simplifying x = -4.866025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.133974596, -4.866025404}
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