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Simplifying 4x2 + 24x + 27 = -6 Reorder the terms: 27 + 24x + 4x2 = -6 Solving 27 + 24x + 4x2 = -6 Solving for variable 'x'. Reorder the terms: 27 + 6 + 24x + 4x2 = -6 + 6 Combine like terms: 27 + 6 = 33 33 + 24x + 4x2 = -6 + 6 Combine like terms: -6 + 6 = 0 33 + 24x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 8.25 + 6x + x2 = 0 Move the constant term to the right: Add '-8.25' to each side of the equation. 8.25 + 6x + -8.25 + x2 = 0 + -8.25 Reorder the terms: 8.25 + -8.25 + 6x + x2 = 0 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + 6x + x2 = 0 + -8.25 6x + x2 = 0 + -8.25 Combine like terms: 0 + -8.25 = -8.25 6x + x2 = -8.25 The x term is 6x. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6x + 9 + x2 = -8.25 + 9 Reorder the terms: 9 + 6x + x2 = -8.25 + 9 Combine like terms: -8.25 + 9 = 0.75 9 + 6x + x2 = 0.75 Factor a perfect square on the left side: (x + 3)(x + 3) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 3) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 3 = 0.866025404 Simplifying x + 3 = 0.866025404 Reorder the terms: 3 + x = 0.866025404 Solving 3 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 0.866025404 + -3 Combine like terms: 3 + -3 = 0 0 + x = 0.866025404 + -3 x = 0.866025404 + -3 Combine like terms: 0.866025404 + -3 = -2.133974596 x = -2.133974596 Simplifying x = -2.133974596Subproblem 2
x + 3 = -0.866025404 Simplifying x + 3 = -0.866025404 Reorder the terms: 3 + x = -0.866025404 Solving 3 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -0.866025404 + -3 Combine like terms: 3 + -3 = 0 0 + x = -0.866025404 + -3 x = -0.866025404 + -3 Combine like terms: -0.866025404 + -3 = -3.866025404 x = -3.866025404 Simplifying x = -3.866025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.133974596, -3.866025404}
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