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Simplifying 6(x + 1)(x + 2) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) Reorder the terms: 6(1 + x)(x + 2) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) Reorder the terms: 6(1 + x)(2 + x) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) Multiply (1 + x) * (2 + x) 6(1(2 + x) + x(2 + x)) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) 6((2 * 1 + x * 1) + x(2 + x)) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) 6((2 + 1x) + x(2 + x)) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) 6(2 + 1x + (2 * x + x * x)) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) 6(2 + 1x + (2x + x2)) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) Combine like terms: 1x + 2x = 3x 6(2 + 3x + x2) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) (2 * 6 + 3x * 6 + x2 * 6) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) (12 + 18x + 6x2) + -6(x + -5)(x + 3) = 17(x + -5)(x + 1) Reorder the terms: 12 + 18x + 6x2 + -6(-5 + x)(x + 3) = 17(x + -5)(x + 1) Reorder the terms: 12 + 18x + 6x2 + -6(-5 + x)(3 + x) = 17(x + -5)(x + 1) Multiply (-5 + x) * (3 + x) 12 + 18x + 6x2 + -6(-5(3 + x) + x(3 + x)) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + -6((3 * -5 + x * -5) + x(3 + x)) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + -6((-15 + -5x) + x(3 + x)) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + -6(-15 + -5x + (3 * x + x * x)) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + -6(-15 + -5x + (3x + x2)) = 17(x + -5)(x + 1) Combine like terms: -5x + 3x = -2x 12 + 18x + 6x2 + -6(-15 + -2x + x2) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + (-15 * -6 + -2x * -6 + x2 * -6) = 17(x + -5)(x + 1) 12 + 18x + 6x2 + (90 + 12x + -6x2) = 17(x + -5)(x + 1) Reorder the terms: 12 + 90 + 18x + 12x + 6x2 + -6x2 = 17(x + -5)(x + 1) Combine like terms: 12 + 90 = 102 102 + 18x + 12x + 6x2 + -6x2 = 17(x + -5)(x + 1) Combine like terms: 18x + 12x = 30x 102 + 30x + 6x2 + -6x2 = 17(x + -5)(x + 1) Combine like terms: 6x2 + -6x2 = 0 102 + 30x + 0 = 17(x + -5)(x + 1) 102 + 30x = 17(x + -5)(x + 1) Reorder the terms: 102 + 30x = 17(-5 + x)(x + 1) Reorder the terms: 102 + 30x = 17(-5 + x)(1 + x) Multiply (-5 + x) * (1 + x) 102 + 30x = 17(-5(1 + x) + x(1 + x)) 102 + 30x = 17((1 * -5 + x * -5) + x(1 + x)) 102 + 30x = 17((-5 + -5x) + x(1 + x)) 102 + 30x = 17(-5 + -5x + (1 * x + x * x)) 102 + 30x = 17(-5 + -5x + (1x + x2)) Combine like terms: -5x + 1x = -4x 102 + 30x = 17(-5 + -4x + x2) 102 + 30x = (-5 * 17 + -4x * 17 + x2 * 17) 102 + 30x = (-85 + -68x + 17x2) Solving 102 + 30x = -85 + -68x + 17x2 Solving for variable 'x'. Reorder the terms: 102 + 85 + 30x + 68x + -17x2 = -85 + -68x + 17x2 + 85 + 68x + -17x2 Combine like terms: 102 + 85 = 187 187 + 30x + 68x + -17x2 = -85 + -68x + 17x2 + 85 + 68x + -17x2 Combine like terms: 30x + 68x = 98x 187 + 98x + -17x2 = -85 + -68x + 17x2 + 85 + 68x + -17x2 Reorder the terms: 187 + 98x + -17x2 = -85 + 85 + -68x + 68x + 17x2 + -17x2 Combine like terms: -85 + 85 = 0 187 + 98x + -17x2 = 0 + -68x + 68x + 17x2 + -17x2 187 + 98x + -17x2 = -68x + 68x + 17x2 + -17x2 Combine like terms: -68x + 68x = 0 187 + 98x + -17x2 = 0 + 17x2 + -17x2 187 + 98x + -17x2 = 17x2 + -17x2 Combine like terms: 17x2 + -17x2 = 0 187 + 98x + -17x2 = 0 Begin completing the square. Divide all terms by -17 the coefficient of the squared term: Divide each side by '-17'. -11 + -5.764705882x + x2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + -5.764705882x + 11 + x2 = 0 + 11 Reorder the terms: -11 + 11 + -5.764705882x + x2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + -5.764705882x + x2 = 0 + 11 -5.764705882x + x2 = 0 + 11 Combine like terms: 0 + 11 = 11 -5.764705882x + x2 = 11 The x term is -5.764705882x. Take half its coefficient (-2.882352941). Square it (8.307958476) and add it to both sides. Add '8.307958476' to each side of the equation. -5.764705882x + 8.307958476 + x2 = 11 + 8.307958476 Reorder the terms: 8.307958476 + -5.764705882x + x2 = 11 + 8.307958476 Combine like terms: 11 + 8.307958476 = 19.307958476 8.307958476 + -5.764705882x + x2 = 19.307958476 Factor a perfect square on the left side: (x + -2.882352941)(x + -2.882352941) = 19.307958476 Calculate the square root of the right side: 4.394082211 Break this problem into two subproblems by setting (x + -2.882352941) equal to 4.394082211 and -4.394082211.Subproblem 1
x + -2.882352941 = 4.394082211 Simplifying x + -2.882352941 = 4.394082211 Reorder the terms: -2.882352941 + x = 4.394082211 Solving -2.882352941 + x = 4.394082211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.882352941' to each side of the equation. -2.882352941 + 2.882352941 + x = 4.394082211 + 2.882352941 Combine like terms: -2.882352941 + 2.882352941 = 0.000000000 0.000000000 + x = 4.394082211 + 2.882352941 x = 4.394082211 + 2.882352941 Combine like terms: 4.394082211 + 2.882352941 = 7.276435152 x = 7.276435152 Simplifying x = 7.276435152Subproblem 2
x + -2.882352941 = -4.394082211 Simplifying x + -2.882352941 = -4.394082211 Reorder the terms: -2.882352941 + x = -4.394082211 Solving -2.882352941 + x = -4.394082211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.882352941' to each side of the equation. -2.882352941 + 2.882352941 + x = -4.394082211 + 2.882352941 Combine like terms: -2.882352941 + 2.882352941 = 0.000000000 0.000000000 + x = -4.394082211 + 2.882352941 x = -4.394082211 + 2.882352941 Combine like terms: -4.394082211 + 2.882352941 = -1.51172927 x = -1.51172927 Simplifying x = -1.51172927Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.276435152, -1.51172927}
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