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Simplifying 32x2 + -3x + -14 = 4x + 1 Reorder the terms: -14 + -3x + 32x2 = 4x + 1 Reorder the terms: -14 + -3x + 32x2 = 1 + 4x Solving -14 + -3x + 32x2 = 1 + 4x Solving for variable 'x'. Reorder the terms: -14 + -1 + -3x + -4x + 32x2 = 1 + 4x + -1 + -4x Combine like terms: -14 + -1 = -15 -15 + -3x + -4x + 32x2 = 1 + 4x + -1 + -4x Combine like terms: -3x + -4x = -7x -15 + -7x + 32x2 = 1 + 4x + -1 + -4x Reorder the terms: -15 + -7x + 32x2 = 1 + -1 + 4x + -4x Combine like terms: 1 + -1 = 0 -15 + -7x + 32x2 = 0 + 4x + -4x -15 + -7x + 32x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -15 + -7x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.46875 + -0.21875x + x2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + -0.21875x + 0.46875 + x2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + -0.21875x + x2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + -0.21875x + x2 = 0 + 0.46875 -0.21875x + x2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 -0.21875x + x2 = 0.46875 The x term is -0.21875x. Take half its coefficient (-0.109375). Square it (0.01196289063) and add it to both sides. Add '0.01196289063' to each side of the equation. -0.21875x + 0.01196289063 + x2 = 0.46875 + 0.01196289063 Reorder the terms: 0.01196289063 + -0.21875x + x2 = 0.46875 + 0.01196289063 Combine like terms: 0.46875 + 0.01196289063 = 0.48071289063 0.01196289063 + -0.21875x + x2 = 0.48071289063 Factor a perfect square on the left side: (x + -0.109375)(x + -0.109375) = 0.48071289063 Calculate the square root of the right side: 0.693334617 Break this problem into two subproblems by setting (x + -0.109375) equal to 0.693334617 and -0.693334617.Subproblem 1
x + -0.109375 = 0.693334617 Simplifying x + -0.109375 = 0.693334617 Reorder the terms: -0.109375 + x = 0.693334617 Solving -0.109375 + x = 0.693334617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.109375' to each side of the equation. -0.109375 + 0.109375 + x = 0.693334617 + 0.109375 Combine like terms: -0.109375 + 0.109375 = 0.000000 0.000000 + x = 0.693334617 + 0.109375 x = 0.693334617 + 0.109375 Combine like terms: 0.693334617 + 0.109375 = 0.802709617 x = 0.802709617 Simplifying x = 0.802709617Subproblem 2
x + -0.109375 = -0.693334617 Simplifying x + -0.109375 = -0.693334617 Reorder the terms: -0.109375 + x = -0.693334617 Solving -0.109375 + x = -0.693334617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.109375' to each side of the equation. -0.109375 + 0.109375 + x = -0.693334617 + 0.109375 Combine like terms: -0.109375 + 0.109375 = 0.000000 0.000000 + x = -0.693334617 + 0.109375 x = -0.693334617 + 0.109375 Combine like terms: -0.693334617 + 0.109375 = -0.583959617 x = -0.583959617 Simplifying x = -0.583959617Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.802709617, -0.583959617}
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