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Simplifying (3x + 5)(x + -2) + (5x + -8)(2x + -3) = 0 Reorder the terms: (5 + 3x)(x + -2) + (5x + -8)(2x + -3) = 0 Reorder the terms: (5 + 3x)(-2 + x) + (5x + -8)(2x + -3) = 0 Multiply (5 + 3x) * (-2 + x) (5(-2 + x) + 3x * (-2 + x)) + (5x + -8)(2x + -3) = 0 ((-2 * 5 + x * 5) + 3x * (-2 + x)) + (5x + -8)(2x + -3) = 0 ((-10 + 5x) + 3x * (-2 + x)) + (5x + -8)(2x + -3) = 0 (-10 + 5x + (-2 * 3x + x * 3x)) + (5x + -8)(2x + -3) = 0 (-10 + 5x + (-6x + 3x2)) + (5x + -8)(2x + -3) = 0 Combine like terms: 5x + -6x = -1x (-10 + -1x + 3x2) + (5x + -8)(2x + -3) = 0 Reorder the terms: -10 + -1x + 3x2 + (-8 + 5x)(2x + -3) = 0 Reorder the terms: -10 + -1x + 3x2 + (-8 + 5x)(-3 + 2x) = 0 Multiply (-8 + 5x) * (-3 + 2x) -10 + -1x + 3x2 + (-8(-3 + 2x) + 5x * (-3 + 2x)) = 0 -10 + -1x + 3x2 + ((-3 * -8 + 2x * -8) + 5x * (-3 + 2x)) = 0 -10 + -1x + 3x2 + ((24 + -16x) + 5x * (-3 + 2x)) = 0 -10 + -1x + 3x2 + (24 + -16x + (-3 * 5x + 2x * 5x)) = 0 -10 + -1x + 3x2 + (24 + -16x + (-15x + 10x2)) = 0 Combine like terms: -16x + -15x = -31x -10 + -1x + 3x2 + (24 + -31x + 10x2) = 0 Reorder the terms: -10 + 24 + -1x + -31x + 3x2 + 10x2 = 0 Combine like terms: -10 + 24 = 14 14 + -1x + -31x + 3x2 + 10x2 = 0 Combine like terms: -1x + -31x = -32x 14 + -32x + 3x2 + 10x2 = 0 Combine like terms: 3x2 + 10x2 = 13x2 14 + -32x + 13x2 = 0 Solving 14 + -32x + 13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 1.076923077 + -2.461538462x + x2 = 0 Move the constant term to the right: Add '-1.076923077' to each side of the equation. 1.076923077 + -2.461538462x + -1.076923077 + x2 = 0 + -1.076923077 Reorder the terms: 1.076923077 + -1.076923077 + -2.461538462x + x2 = 0 + -1.076923077 Combine like terms: 1.076923077 + -1.076923077 = 0.000000000 0.000000000 + -2.461538462x + x2 = 0 + -1.076923077 -2.461538462x + x2 = 0 + -1.076923077 Combine like terms: 0 + -1.076923077 = -1.076923077 -2.461538462x + x2 = -1.076923077 The x term is -2.461538462x. Take half its coefficient (-1.230769231). Square it (1.514792900) and add it to both sides. Add '1.514792900' to each side of the equation. -2.461538462x + 1.514792900 + x2 = -1.076923077 + 1.514792900 Reorder the terms: 1.514792900 + -2.461538462x + x2 = -1.076923077 + 1.514792900 Combine like terms: -1.076923077 + 1.514792900 = 0.437869823 1.514792900 + -2.461538462x + x2 = 0.437869823 Factor a perfect square on the left side: (x + -1.230769231)(x + -1.230769231) = 0.437869823 Calculate the square root of the right side: 0.661717329 Break this problem into two subproblems by setting (x + -1.230769231) equal to 0.661717329 and -0.661717329.Subproblem 1
x + -1.230769231 = 0.661717329 Simplifying x + -1.230769231 = 0.661717329 Reorder the terms: -1.230769231 + x = 0.661717329 Solving -1.230769231 + x = 0.661717329 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.230769231' to each side of the equation. -1.230769231 + 1.230769231 + x = 0.661717329 + 1.230769231 Combine like terms: -1.230769231 + 1.230769231 = 0.000000000 0.000000000 + x = 0.661717329 + 1.230769231 x = 0.661717329 + 1.230769231 Combine like terms: 0.661717329 + 1.230769231 = 1.89248656 x = 1.89248656 Simplifying x = 1.89248656Subproblem 2
x + -1.230769231 = -0.661717329 Simplifying x + -1.230769231 = -0.661717329 Reorder the terms: -1.230769231 + x = -0.661717329 Solving -1.230769231 + x = -0.661717329 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.230769231' to each side of the equation. -1.230769231 + 1.230769231 + x = -0.661717329 + 1.230769231 Combine like terms: -1.230769231 + 1.230769231 = 0.000000000 0.000000000 + x = -0.661717329 + 1.230769231 x = -0.661717329 + 1.230769231 Combine like terms: -0.661717329 + 1.230769231 = 0.569051902 x = 0.569051902 Simplifying x = 0.569051902Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.89248656, 0.569051902}
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