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Simplifying x(x + 4) + x(x + 2) = 2x * 2x + 1 Reorder the terms: x(4 + x) + x(x + 2) = 2x * 2x + 1 (4 * x + x * x) + x(x + 2) = 2x * 2x + 1 (4x + x2) + x(x + 2) = 2x * 2x + 1 Reorder the terms: 4x + x2 + x(2 + x) = 2x * 2x + 1 4x + x2 + (2 * x + x * x) = 2x * 2x + 1 4x + x2 + (2x + x2) = 2x * 2x + 1 Reorder the terms: 4x + 2x + x2 + x2 = 2x * 2x + 1 Combine like terms: 4x + 2x = 6x 6x + x2 + x2 = 2x * 2x + 1 Combine like terms: x2 + x2 = 2x2 6x + 2x2 = 2x * 2x + 1 Reorder the terms for easier multiplication: 6x + 2x2 = 2 * 2x * x + 1 Multiply 2 * 2 6x + 2x2 = 4x * x + 1 Multiply x * x 6x + 2x2 = 4x2 + 1 Reorder the terms: 6x + 2x2 = 1 + 4x2 Solving 6x + 2x2 = 1 + 4x2 Solving for variable 'x'. Reorder the terms: -1 + 6x + 2x2 + -4x2 = 1 + 4x2 + -1 + -4x2 Combine like terms: 2x2 + -4x2 = -2x2 -1 + 6x + -2x2 = 1 + 4x2 + -1 + -4x2 Reorder the terms: -1 + 6x + -2x2 = 1 + -1 + 4x2 + -4x2 Combine like terms: 1 + -1 = 0 -1 + 6x + -2x2 = 0 + 4x2 + -4x2 -1 + 6x + -2x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 -1 + 6x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -3x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -3x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -3x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -3x + x2 = 0 + -0.5 -3x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -3x + x2 = -0.5 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = -0.5 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -0.5 + 2.25 Combine like terms: -0.5 + 2.25 = 1.75 2.25 + -3x + x2 = 1.75 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + -1.5) equal to 1.322875656 and -1.322875656.Subproblem 1
x + -1.5 = 1.322875656 Simplifying x + -1.5 = 1.322875656 Reorder the terms: -1.5 + x = 1.322875656 Solving -1.5 + x = 1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.322875656 + 1.5 x = 1.322875656 + 1.5 Combine like terms: 1.322875656 + 1.5 = 2.822875656 x = 2.822875656 Simplifying x = 2.822875656Subproblem 2
x + -1.5 = -1.322875656 Simplifying x + -1.5 = -1.322875656 Reorder the terms: -1.5 + x = -1.322875656 Solving -1.5 + x = -1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.322875656 + 1.5 x = -1.322875656 + 1.5 Combine like terms: -1.322875656 + 1.5 = 0.177124344 x = 0.177124344 Simplifying x = 0.177124344Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.822875656, 0.177124344}
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