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Simplifying 4x2 + 31x + 23 = 0 Reorder the terms: 23 + 31x + 4x2 = 0 Solving 23 + 31x + 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'. 5.75 + 7.75x + x2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + 7.75x + -5.75 + x2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + 7.75x + x2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + 7.75x + x2 = 0 + -5.75 7.75x + x2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 7.75x + x2 = -5.75 The x term is 7.75x. Take half its coefficient (3.875). Square it (15.015625) and add it to both sides. Add '15.015625' to each side of the equation. 7.75x + 15.015625 + x2 = -5.75 + 15.015625 Reorder the terms: 15.015625 + 7.75x + x2 = -5.75 + 15.015625 Combine like terms: -5.75 + 15.015625 = 9.265625 15.015625 + 7.75x + x2 = 9.265625 Factor a perfect square on the left side: (x + 3.875)(x + 3.875) = 9.265625 Calculate the square root of the right side: 3.043948915 Break this problem into two subproblems by setting (x + 3.875) equal to 3.043948915 and -3.043948915.Subproblem 1
x + 3.875 = 3.043948915 Simplifying x + 3.875 = 3.043948915 Reorder the terms: 3.875 + x = 3.043948915 Solving 3.875 + x = 3.043948915 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = 3.043948915 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = 3.043948915 + -3.875 x = 3.043948915 + -3.875 Combine like terms: 3.043948915 + -3.875 = -0.831051085 x = -0.831051085 Simplifying x = -0.831051085Subproblem 2
x + 3.875 = -3.043948915 Simplifying x + 3.875 = -3.043948915 Reorder the terms: 3.875 + x = -3.043948915 Solving 3.875 + x = -3.043948915 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = -3.043948915 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = -3.043948915 + -3.875 x = -3.043948915 + -3.875 Combine like terms: -3.043948915 + -3.875 = -6.918948915 x = -6.918948915 Simplifying x = -6.918948915Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.831051085, -6.918948915}
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