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Simplifying (6x + -10)(3x + 8) + -1(7 + -4x)(5 + 5x) = 0 Reorder the terms: (-10 + 6x)(3x + 8) + -1(7 + -4x)(5 + 5x) = 0 Reorder the terms: (-10 + 6x)(8 + 3x) + -1(7 + -4x)(5 + 5x) = 0 Multiply (-10 + 6x) * (8 + 3x) (-10(8 + 3x) + 6x * (8 + 3x)) + -1(7 + -4x)(5 + 5x) = 0 ((8 * -10 + 3x * -10) + 6x * (8 + 3x)) + -1(7 + -4x)(5 + 5x) = 0 ((-80 + -30x) + 6x * (8 + 3x)) + -1(7 + -4x)(5 + 5x) = 0 (-80 + -30x + (8 * 6x + 3x * 6x)) + -1(7 + -4x)(5 + 5x) = 0 (-80 + -30x + (48x + 18x2)) + -1(7 + -4x)(5 + 5x) = 0 Combine like terms: -30x + 48x = 18x (-80 + 18x + 18x2) + -1(7 + -4x)(5 + 5x) = 0 Multiply (7 + -4x) * (5 + 5x) -80 + 18x + 18x2 + -1(7(5 + 5x) + -4x * (5 + 5x)) = 0 -80 + 18x + 18x2 + -1((5 * 7 + 5x * 7) + -4x * (5 + 5x)) = 0 -80 + 18x + 18x2 + -1((35 + 35x) + -4x * (5 + 5x)) = 0 -80 + 18x + 18x2 + -1(35 + 35x + (5 * -4x + 5x * -4x)) = 0 -80 + 18x + 18x2 + -1(35 + 35x + (-20x + -20x2)) = 0 Combine like terms: 35x + -20x = 15x -80 + 18x + 18x2 + -1(35 + 15x + -20x2) = 0 -80 + 18x + 18x2 + (35 * -1 + 15x * -1 + -20x2 * -1) = 0 -80 + 18x + 18x2 + (-35 + -15x + 20x2) = 0 Reorder the terms: -80 + -35 + 18x + -15x + 18x2 + 20x2 = 0 Combine like terms: -80 + -35 = -115 -115 + 18x + -15x + 18x2 + 20x2 = 0 Combine like terms: 18x + -15x = 3x -115 + 3x + 18x2 + 20x2 = 0 Combine like terms: 18x2 + 20x2 = 38x2 -115 + 3x + 38x2 = 0 Solving -115 + 3x + 38x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 38 the coefficient of the squared term: Divide each side by '38'. -3.026315789 + 0.07894736842x + x2 = 0 Move the constant term to the right: Add '3.026315789' to each side of the equation. -3.026315789 + 0.07894736842x + 3.026315789 + x2 = 0 + 3.026315789 Reorder the terms: -3.026315789 + 3.026315789 + 0.07894736842x + x2 = 0 + 3.026315789 Combine like terms: -3.026315789 + 3.026315789 = 0.000000000 0.000000000 + 0.07894736842x + x2 = 0 + 3.026315789 0.07894736842x + x2 = 0 + 3.026315789 Combine like terms: 0 + 3.026315789 = 3.026315789 0.07894736842x + x2 = 3.026315789 The x term is 0.07894736842x. Take half its coefficient (0.03947368421). Square it (0.001558171745) and add it to both sides. Add '0.001558171745' to each side of the equation. 0.07894736842x + 0.001558171745 + x2 = 3.026315789 + 0.001558171745 Reorder the terms: 0.001558171745 + 0.07894736842x + x2 = 3.026315789 + 0.001558171745 Combine like terms: 3.026315789 + 0.001558171745 = 3.027873960745 0.001558171745 + 0.07894736842x + x2 = 3.027873960745 Factor a perfect square on the left side: (x + 0.03947368421)(x + 0.03947368421) = 3.027873960745 Calculate the square root of the right side: 1.740078723 Break this problem into two subproblems by setting (x + 0.03947368421) equal to 1.740078723 and -1.740078723.Subproblem 1
x + 0.03947368421 = 1.740078723 Simplifying x + 0.03947368421 = 1.740078723 Reorder the terms: 0.03947368421 + x = 1.740078723 Solving 0.03947368421 + x = 1.740078723 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03947368421' to each side of the equation. 0.03947368421 + -0.03947368421 + x = 1.740078723 + -0.03947368421 Combine like terms: 0.03947368421 + -0.03947368421 = 0.00000000000 0.00000000000 + x = 1.740078723 + -0.03947368421 x = 1.740078723 + -0.03947368421 Combine like terms: 1.740078723 + -0.03947368421 = 1.70060503879 x = 1.70060503879 Simplifying x = 1.70060503879Subproblem 2
x + 0.03947368421 = -1.740078723 Simplifying x + 0.03947368421 = -1.740078723 Reorder the terms: 0.03947368421 + x = -1.740078723 Solving 0.03947368421 + x = -1.740078723 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03947368421' to each side of the equation. 0.03947368421 + -0.03947368421 + x = -1.740078723 + -0.03947368421 Combine like terms: 0.03947368421 + -0.03947368421 = 0.00000000000 0.00000000000 + x = -1.740078723 + -0.03947368421 x = -1.740078723 + -0.03947368421 Combine like terms: -1.740078723 + -0.03947368421 = -1.77955240721 x = -1.77955240721 Simplifying x = -1.77955240721Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.70060503879, -1.77955240721}
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