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Simplifying 4x2 + 7x + -27 = 0 Reorder the terms: -27 + 7x + 4x2 = 0 Solving -27 + 7x + 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'. -6.75 + 1.75x + x2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + 1.75x + 6.75 + x2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + 1.75x + x2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + 1.75x + x2 = 0 + 6.75 1.75x + x2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 1.75x + x2 = 6.75 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = 6.75 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = 6.75 + 0.765625 Combine like terms: 6.75 + 0.765625 = 7.515625 0.765625 + 1.75x + x2 = 7.515625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 7.515625 Calculate the square root of the right side: 2.741464025 Break this problem into two subproblems by setting (x + 0.875) equal to 2.741464025 and -2.741464025.Subproblem 1
x + 0.875 = 2.741464025 Simplifying x + 0.875 = 2.741464025 Reorder the terms: 0.875 + x = 2.741464025 Solving 0.875 + x = 2.741464025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 2.741464025 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 2.741464025 + -0.875 x = 2.741464025 + -0.875 Combine like terms: 2.741464025 + -0.875 = 1.866464025 x = 1.866464025 Simplifying x = 1.866464025Subproblem 2
x + 0.875 = -2.741464025 Simplifying x + 0.875 = -2.741464025 Reorder the terms: 0.875 + x = -2.741464025 Solving 0.875 + x = -2.741464025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -2.741464025 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -2.741464025 + -0.875 x = -2.741464025 + -0.875 Combine like terms: -2.741464025 + -0.875 = -3.616464025 x = -3.616464025 Simplifying x = -3.616464025Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.866464025, -3.616464025}
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