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Simplifying 4x2 + -31x + 47 = 3x + 2 Reorder the terms: 47 + -31x + 4x2 = 3x + 2 Reorder the terms: 47 + -31x + 4x2 = 2 + 3x Solving 47 + -31x + 4x2 = 2 + 3x Solving for variable 'x'. Reorder the terms: 47 + -2 + -31x + -3x + 4x2 = 2 + 3x + -2 + -3x Combine like terms: 47 + -2 = 45 45 + -31x + -3x + 4x2 = 2 + 3x + -2 + -3x Combine like terms: -31x + -3x = -34x 45 + -34x + 4x2 = 2 + 3x + -2 + -3x Reorder the terms: 45 + -34x + 4x2 = 2 + -2 + 3x + -3x Combine like terms: 2 + -2 = 0 45 + -34x + 4x2 = 0 + 3x + -3x 45 + -34x + 4x2 = 3x + -3x Combine like terms: 3x + -3x = 0 45 + -34x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 11.25 + -8.5x + x2 = 0 Move the constant term to the right: Add '-11.25' to each side of the equation. 11.25 + -8.5x + -11.25 + x2 = 0 + -11.25 Reorder the terms: 11.25 + -11.25 + -8.5x + x2 = 0 + -11.25 Combine like terms: 11.25 + -11.25 = 0.00 0.00 + -8.5x + x2 = 0 + -11.25 -8.5x + x2 = 0 + -11.25 Combine like terms: 0 + -11.25 = -11.25 -8.5x + x2 = -11.25 The x term is -8.5x. Take half its coefficient (-4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. -8.5x + 18.0625 + x2 = -11.25 + 18.0625 Reorder the terms: 18.0625 + -8.5x + x2 = -11.25 + 18.0625 Combine like terms: -11.25 + 18.0625 = 6.8125 18.0625 + -8.5x + x2 = 6.8125 Factor a perfect square on the left side: (x + -4.25)(x + -4.25) = 6.8125 Calculate the square root of the right side: 2.610076627 Break this problem into two subproblems by setting (x + -4.25) equal to 2.610076627 and -2.610076627.Subproblem 1
x + -4.25 = 2.610076627 Simplifying x + -4.25 = 2.610076627 Reorder the terms: -4.25 + x = 2.610076627 Solving -4.25 + x = 2.610076627 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.25' to each side of the equation. -4.25 + 4.25 + x = 2.610076627 + 4.25 Combine like terms: -4.25 + 4.25 = 0.00 0.00 + x = 2.610076627 + 4.25 x = 2.610076627 + 4.25 Combine like terms: 2.610076627 + 4.25 = 6.860076627 x = 6.860076627 Simplifying x = 6.860076627Subproblem 2
x + -4.25 = -2.610076627 Simplifying x + -4.25 = -2.610076627 Reorder the terms: -4.25 + x = -2.610076627 Solving -4.25 + x = -2.610076627 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.25' to each side of the equation. -4.25 + 4.25 + x = -2.610076627 + 4.25 Combine like terms: -4.25 + 4.25 = 0.00 0.00 + x = -2.610076627 + 4.25 x = -2.610076627 + 4.25 Combine like terms: -2.610076627 + 4.25 = 1.639923373 x = 1.639923373 Simplifying x = 1.639923373Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.860076627, 1.639923373}
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