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Simplifying 32 + -32x + 8x2 + -64 + 16x + -1x2 = 0 Reorder the terms: 32 + -64 + -32x + 16x + 8x2 + -1x2 = 0 Combine like terms: 32 + -64 = -32 -32 + -32x + 16x + 8x2 + -1x2 = 0 Combine like terms: -32x + 16x = -16x -32 + -16x + 8x2 + -1x2 = 0 Combine like terms: 8x2 + -1x2 = 7x2 -32 + -16x + 7x2 = 0 Solving -32 + -16x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.571428571 + -2.285714286x + x2 = 0 Move the constant term to the right: Add '4.571428571' to each side of the equation. -4.571428571 + -2.285714286x + 4.571428571 + x2 = 0 + 4.571428571 Reorder the terms: -4.571428571 + 4.571428571 + -2.285714286x + x2 = 0 + 4.571428571 Combine like terms: -4.571428571 + 4.571428571 = 0.000000000 0.000000000 + -2.285714286x + x2 = 0 + 4.571428571 -2.285714286x + x2 = 0 + 4.571428571 Combine like terms: 0 + 4.571428571 = 4.571428571 -2.285714286x + x2 = 4.571428571 The x term is -2.285714286x. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286x + 1.306122449 + x2 = 4.571428571 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286x + x2 = 4.571428571 + 1.306122449 Combine like terms: 4.571428571 + 1.306122449 = 5.87755102 1.306122449 + -2.285714286x + x2 = 5.87755102 Factor a perfect square on the left side: (x + -1.142857143)(x + -1.142857143) = 5.87755102 Calculate the square root of the right side: 2.424366107 Break this problem into two subproblems by setting (x + -1.142857143) equal to 2.424366107 and -2.424366107.Subproblem 1
x + -1.142857143 = 2.424366107 Simplifying x + -1.142857143 = 2.424366107 Reorder the terms: -1.142857143 + x = 2.424366107 Solving -1.142857143 + x = 2.424366107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = 2.424366107 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = 2.424366107 + 1.142857143 x = 2.424366107 + 1.142857143 Combine like terms: 2.424366107 + 1.142857143 = 3.56722325 x = 3.56722325 Simplifying x = 3.56722325Subproblem 2
x + -1.142857143 = -2.424366107 Simplifying x + -1.142857143 = -2.424366107 Reorder the terms: -1.142857143 + x = -2.424366107 Solving -1.142857143 + x = -2.424366107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = -2.424366107 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = -2.424366107 + 1.142857143 x = -2.424366107 + 1.142857143 Combine like terms: -2.424366107 + 1.142857143 = -1.281508964 x = -1.281508964 Simplifying x = -1.281508964Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.56722325, -1.281508964}
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