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Simplifying (2x + -1)(x + 2) + (3x + 1)(x + -1) = 0 Reorder the terms: (-1 + 2x)(x + 2) + (3x + 1)(x + -1) = 0 Reorder the terms: (-1 + 2x)(2 + x) + (3x + 1)(x + -1) = 0 Multiply (-1 + 2x) * (2 + x) (-1(2 + x) + 2x * (2 + x)) + (3x + 1)(x + -1) = 0 ((2 * -1 + x * -1) + 2x * (2 + x)) + (3x + 1)(x + -1) = 0 ((-2 + -1x) + 2x * (2 + x)) + (3x + 1)(x + -1) = 0 (-2 + -1x + (2 * 2x + x * 2x)) + (3x + 1)(x + -1) = 0 (-2 + -1x + (4x + 2x2)) + (3x + 1)(x + -1) = 0 Combine like terms: -1x + 4x = 3x (-2 + 3x + 2x2) + (3x + 1)(x + -1) = 0 Reorder the terms: -2 + 3x + 2x2 + (1 + 3x)(x + -1) = 0 Reorder the terms: -2 + 3x + 2x2 + (1 + 3x)(-1 + x) = 0 Multiply (1 + 3x) * (-1 + x) -2 + 3x + 2x2 + (1(-1 + x) + 3x * (-1 + x)) = 0 -2 + 3x + 2x2 + ((-1 * 1 + x * 1) + 3x * (-1 + x)) = 0 -2 + 3x + 2x2 + ((-1 + 1x) + 3x * (-1 + x)) = 0 -2 + 3x + 2x2 + (-1 + 1x + (-1 * 3x + x * 3x)) = 0 -2 + 3x + 2x2 + (-1 + 1x + (-3x + 3x2)) = 0 Combine like terms: 1x + -3x = -2x -2 + 3x + 2x2 + (-1 + -2x + 3x2) = 0 Reorder the terms: -2 + -1 + 3x + -2x + 2x2 + 3x2 = 0 Combine like terms: -2 + -1 = -3 -3 + 3x + -2x + 2x2 + 3x2 = 0 Combine like terms: 3x + -2x = 1x -3 + 1x + 2x2 + 3x2 = 0 Combine like terms: 2x2 + 3x2 = 5x2 -3 + 1x + 5x2 = 0 Solving -3 + 1x + 5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + 0.2x + x2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + 0.2x + 0.6 + x2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + 0.2x + x2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + 0.2x + x2 = 0 + 0.6 0.2x + x2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 0.2x + x2 = 0.6 The x term is 0.2x. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2x + 0.01 + x2 = 0.6 + 0.01 Reorder the terms: 0.01 + 0.2x + x2 = 0.6 + 0.01 Combine like terms: 0.6 + 0.01 = 0.61 0.01 + 0.2x + x2 = 0.61 Factor a perfect square on the left side: (x + 0.1)(x + 0.1) = 0.61 Calculate the square root of the right side: 0.781024968 Break this problem into two subproblems by setting (x + 0.1) equal to 0.781024968 and -0.781024968.Subproblem 1
x + 0.1 = 0.781024968 Simplifying x + 0.1 = 0.781024968 Reorder the terms: 0.1 + x = 0.781024968 Solving 0.1 + x = 0.781024968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 0.781024968 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 0.781024968 + -0.1 x = 0.781024968 + -0.1 Combine like terms: 0.781024968 + -0.1 = 0.681024968 x = 0.681024968 Simplifying x = 0.681024968Subproblem 2
x + 0.1 = -0.781024968 Simplifying x + 0.1 = -0.781024968 Reorder the terms: 0.1 + x = -0.781024968 Solving 0.1 + x = -0.781024968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -0.781024968 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -0.781024968 + -0.1 x = -0.781024968 + -0.1 Combine like terms: -0.781024968 + -0.1 = -0.881024968 x = -0.881024968 Simplifying x = -0.881024968Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.681024968, -0.881024968}
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