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Simplifying (2x + -1)(x + -4) + -1(3x + 2)(3x + -1) = 0 Reorder the terms: (-1 + 2x)(x + -4) + -1(3x + 2)(3x + -1) = 0 Reorder the terms: (-1 + 2x)(-4 + x) + -1(3x + 2)(3x + -1) = 0 Multiply (-1 + 2x) * (-4 + x) (-1(-4 + x) + 2x * (-4 + x)) + -1(3x + 2)(3x + -1) = 0 ((-4 * -1 + x * -1) + 2x * (-4 + x)) + -1(3x + 2)(3x + -1) = 0 ((4 + -1x) + 2x * (-4 + x)) + -1(3x + 2)(3x + -1) = 0 (4 + -1x + (-4 * 2x + x * 2x)) + -1(3x + 2)(3x + -1) = 0 (4 + -1x + (-8x + 2x2)) + -1(3x + 2)(3x + -1) = 0 Combine like terms: -1x + -8x = -9x (4 + -9x + 2x2) + -1(3x + 2)(3x + -1) = 0 Reorder the terms: 4 + -9x + 2x2 + -1(2 + 3x)(3x + -1) = 0 Reorder the terms: 4 + -9x + 2x2 + -1(2 + 3x)(-1 + 3x) = 0 Multiply (2 + 3x) * (-1 + 3x) 4 + -9x + 2x2 + -1(2(-1 + 3x) + 3x * (-1 + 3x)) = 0 4 + -9x + 2x2 + -1((-1 * 2 + 3x * 2) + 3x * (-1 + 3x)) = 0 4 + -9x + 2x2 + -1((-2 + 6x) + 3x * (-1 + 3x)) = 0 4 + -9x + 2x2 + -1(-2 + 6x + (-1 * 3x + 3x * 3x)) = 0 4 + -9x + 2x2 + -1(-2 + 6x + (-3x + 9x2)) = 0 Combine like terms: 6x + -3x = 3x 4 + -9x + 2x2 + -1(-2 + 3x + 9x2) = 0 4 + -9x + 2x2 + (-2 * -1 + 3x * -1 + 9x2 * -1) = 0 4 + -9x + 2x2 + (2 + -3x + -9x2) = 0 Reorder the terms: 4 + 2 + -9x + -3x + 2x2 + -9x2 = 0 Combine like terms: 4 + 2 = 6 6 + -9x + -3x + 2x2 + -9x2 = 0 Combine like terms: -9x + -3x = -12x 6 + -12x + 2x2 + -9x2 = 0 Combine like terms: 2x2 + -9x2 = -7x2 6 + -12x + -7x2 = 0 Solving 6 + -12x + -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'. -0.8571428571 + 1.714285714x + x2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + 1.714285714x + 0.8571428571 + x2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + 1.714285714x + x2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + 1.714285714x + x2 = 0 + 0.8571428571 1.714285714x + x2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 1.714285714x + x2 = 0.8571428571 The x term is 1.714285714x. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714x + 0.7346938773 + x2 = 0.8571428571 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714x + x2 = 0.8571428571 + 0.7346938773 Combine like terms: 0.8571428571 + 0.7346938773 = 1.5918367344 0.7346938773 + 1.714285714x + x2 = 1.5918367344 Factor a perfect square on the left side: (x + 0.857142857)(x + 0.857142857) = 1.5918367344 Calculate the square root of the right side: 1.261680124 Break this problem into two subproblems by setting (x + 0.857142857) equal to 1.261680124 and -1.261680124.Subproblem 1
x + 0.857142857 = 1.261680124 Simplifying x + 0.857142857 = 1.261680124 Reorder the terms: 0.857142857 + x = 1.261680124 Solving 0.857142857 + x = 1.261680124 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = 1.261680124 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = 1.261680124 + -0.857142857 x = 1.261680124 + -0.857142857 Combine like terms: 1.261680124 + -0.857142857 = 0.404537267 x = 0.404537267 Simplifying x = 0.404537267Subproblem 2
x + 0.857142857 = -1.261680124 Simplifying x + 0.857142857 = -1.261680124 Reorder the terms: 0.857142857 + x = -1.261680124 Solving 0.857142857 + x = -1.261680124 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = -1.261680124 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = -1.261680124 + -0.857142857 x = -1.261680124 + -0.857142857 Combine like terms: -1.261680124 + -0.857142857 = -2.118822981 x = -2.118822981 Simplifying x = -2.118822981Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.404537267, -2.118822981}
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