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Simplifying (4x2 + -4x + -67) + (4x2 + 6x) = 0 Reorder the terms: (-67 + -4x + 4x2) + (4x2 + 6x) = 0 Remove parenthesis around (-67 + -4x + 4x2) -67 + -4x + 4x2 + (4x2 + 6x) = 0 Reorder the terms: -67 + -4x + 4x2 + (6x + 4x2) = 0 Remove parenthesis around (6x + 4x2) -67 + -4x + 4x2 + 6x + 4x2 = 0 Reorder the terms: -67 + -4x + 6x + 4x2 + 4x2 = 0 Combine like terms: -4x + 6x = 2x -67 + 2x + 4x2 + 4x2 = 0 Combine like terms: 4x2 + 4x2 = 8x2 -67 + 2x + 8x2 = 0 Solving -67 + 2x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -8.375 + 0.25x + x2 = 0 Move the constant term to the right: Add '8.375' to each side of the equation. -8.375 + 0.25x + 8.375 + x2 = 0 + 8.375 Reorder the terms: -8.375 + 8.375 + 0.25x + x2 = 0 + 8.375 Combine like terms: -8.375 + 8.375 = 0.000 0.000 + 0.25x + x2 = 0 + 8.375 0.25x + x2 = 0 + 8.375 Combine like terms: 0 + 8.375 = 8.375 0.25x + x2 = 8.375 The x term is 0.25x. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25x + 0.015625 + x2 = 8.375 + 0.015625 Reorder the terms: 0.015625 + 0.25x + x2 = 8.375 + 0.015625 Combine like terms: 8.375 + 0.015625 = 8.390625 0.015625 + 0.25x + x2 = 8.390625 Factor a perfect square on the left side: (x + 0.125)(x + 0.125) = 8.390625 Calculate the square root of the right side: 2.896657557 Break this problem into two subproblems by setting (x + 0.125) equal to 2.896657557 and -2.896657557.Subproblem 1
x + 0.125 = 2.896657557 Simplifying x + 0.125 = 2.896657557 Reorder the terms: 0.125 + x = 2.896657557 Solving 0.125 + x = 2.896657557 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 2.896657557 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 2.896657557 + -0.125 x = 2.896657557 + -0.125 Combine like terms: 2.896657557 + -0.125 = 2.771657557 x = 2.771657557 Simplifying x = 2.771657557Subproblem 2
x + 0.125 = -2.896657557 Simplifying x + 0.125 = -2.896657557 Reorder the terms: 0.125 + x = -2.896657557 Solving 0.125 + x = -2.896657557 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -2.896657557 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -2.896657557 + -0.125 x = -2.896657557 + -0.125 Combine like terms: -2.896657557 + -0.125 = -3.021657557 x = -3.021657557 Simplifying x = -3.021657557Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.771657557, -3.021657557}
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