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Simplifying (x + 8)(3x + -10) + 8x2 + -1(5 + x)(2x + -3) = 0 Reorder the terms: (8 + x)(3x + -10) + 8x2 + -1(5 + x)(2x + -3) = 0 Reorder the terms: (8 + x)(-10 + 3x) + 8x2 + -1(5 + x)(2x + -3) = 0 Multiply (8 + x) * (-10 + 3x) (8(-10 + 3x) + x(-10 + 3x)) + 8x2 + -1(5 + x)(2x + -3) = 0 ((-10 * 8 + 3x * 8) + x(-10 + 3x)) + 8x2 + -1(5 + x)(2x + -3) = 0 ((-80 + 24x) + x(-10 + 3x)) + 8x2 + -1(5 + x)(2x + -3) = 0 (-80 + 24x + (-10 * x + 3x * x)) + 8x2 + -1(5 + x)(2x + -3) = 0 (-80 + 24x + (-10x + 3x2)) + 8x2 + -1(5 + x)(2x + -3) = 0 Combine like terms: 24x + -10x = 14x (-80 + 14x + 3x2) + 8x2 + -1(5 + x)(2x + -3) = 0 Reorder the terms: -80 + 14x + 3x2 + 8x2 + -1(5 + x)(-3 + 2x) = 0 Multiply (5 + x) * (-3 + 2x) -80 + 14x + 3x2 + 8x2 + -1(5(-3 + 2x) + x(-3 + 2x)) = 0 -80 + 14x + 3x2 + 8x2 + -1((-3 * 5 + 2x * 5) + x(-3 + 2x)) = 0 -80 + 14x + 3x2 + 8x2 + -1((-15 + 10x) + x(-3 + 2x)) = 0 -80 + 14x + 3x2 + 8x2 + -1(-15 + 10x + (-3 * x + 2x * x)) = 0 -80 + 14x + 3x2 + 8x2 + -1(-15 + 10x + (-3x + 2x2)) = 0 Combine like terms: 10x + -3x = 7x -80 + 14x + 3x2 + 8x2 + -1(-15 + 7x + 2x2) = 0 -80 + 14x + 3x2 + 8x2 + (-15 * -1 + 7x * -1 + 2x2 * -1) = 0 -80 + 14x + 3x2 + 8x2 + (15 + -7x + -2x2) = 0 Reorder the terms: -80 + 15 + 14x + -7x + 3x2 + 8x2 + -2x2 = 0 Combine like terms: -80 + 15 = -65 -65 + 14x + -7x + 3x2 + 8x2 + -2x2 = 0 Combine like terms: 14x + -7x = 7x -65 + 7x + 3x2 + 8x2 + -2x2 = 0 Combine like terms: 3x2 + 8x2 = 11x2 -65 + 7x + 11x2 + -2x2 = 0 Combine like terms: 11x2 + -2x2 = 9x2 -65 + 7x + 9x2 = 0 Solving -65 + 7x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -7.222222222 + 0.7777777778x + x2 = 0 Move the constant term to the right: Add '7.222222222' to each side of the equation. -7.222222222 + 0.7777777778x + 7.222222222 + x2 = 0 + 7.222222222 Reorder the terms: -7.222222222 + 7.222222222 + 0.7777777778x + x2 = 0 + 7.222222222 Combine like terms: -7.222222222 + 7.222222222 = 0.000000000 0.000000000 + 0.7777777778x + x2 = 0 + 7.222222222 0.7777777778x + x2 = 0 + 7.222222222 Combine like terms: 0 + 7.222222222 = 7.222222222 0.7777777778x + x2 = 7.222222222 The x term is 0.7777777778x. Take half its coefficient (0.3888888889). Square it (0.1512345679) and add it to both sides. Add '0.1512345679' to each side of the equation. 0.7777777778x + 0.1512345679 + x2 = 7.222222222 + 0.1512345679 Reorder the terms: 0.1512345679 + 0.7777777778x + x2 = 7.222222222 + 0.1512345679 Combine like terms: 7.222222222 + 0.1512345679 = 7.3734567899 0.1512345679 + 0.7777777778x + x2 = 7.3734567899 Factor a perfect square on the left side: (x + 0.3888888889)(x + 0.3888888889) = 7.3734567899 Calculate the square root of the right side: 2.71541098 Break this problem into two subproblems by setting (x + 0.3888888889) equal to 2.71541098 and -2.71541098.Subproblem 1
x + 0.3888888889 = 2.71541098 Simplifying x + 0.3888888889 = 2.71541098 Reorder the terms: 0.3888888889 + x = 2.71541098 Solving 0.3888888889 + x = 2.71541098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + x = 2.71541098 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + x = 2.71541098 + -0.3888888889 x = 2.71541098 + -0.3888888889 Combine like terms: 2.71541098 + -0.3888888889 = 2.3265220911 x = 2.3265220911 Simplifying x = 2.3265220911Subproblem 2
x + 0.3888888889 = -2.71541098 Simplifying x + 0.3888888889 = -2.71541098 Reorder the terms: 0.3888888889 + x = -2.71541098 Solving 0.3888888889 + x = -2.71541098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + x = -2.71541098 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + x = -2.71541098 + -0.3888888889 x = -2.71541098 + -0.3888888889 Combine like terms: -2.71541098 + -0.3888888889 = -3.1042998689 x = -3.1042998689 Simplifying x = -3.1042998689Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.3265220911, -3.1042998689}
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