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Simplifying 5(3x + -8) * 3(7x + 6) = 6(8x + -3) + -4(2x + 5) Reorder the terms: 5(-8 + 3x) * 3(7x + 6) = 6(8x + -3) + -4(2x + 5) Reorder the terms: 5(-8 + 3x) * 3(6 + 7x) = 6(8x + -3) + -4(2x + 5) Reorder the terms for easier multiplication: 5 * 3(-8 + 3x)(6 + 7x) = 6(8x + -3) + -4(2x + 5) Multiply 5 * 3 15(-8 + 3x)(6 + 7x) = 6(8x + -3) + -4(2x + 5) Multiply (-8 + 3x) * (6 + 7x) 15(-8(6 + 7x) + 3x * (6 + 7x)) = 6(8x + -3) + -4(2x + 5) 15((6 * -8 + 7x * -8) + 3x * (6 + 7x)) = 6(8x + -3) + -4(2x + 5) 15((-48 + -56x) + 3x * (6 + 7x)) = 6(8x + -3) + -4(2x + 5) 15(-48 + -56x + (6 * 3x + 7x * 3x)) = 6(8x + -3) + -4(2x + 5) 15(-48 + -56x + (18x + 21x2)) = 6(8x + -3) + -4(2x + 5) Combine like terms: -56x + 18x = -38x 15(-48 + -38x + 21x2) = 6(8x + -3) + -4(2x + 5) (-48 * 15 + -38x * 15 + 21x2 * 15) = 6(8x + -3) + -4(2x + 5) (-720 + -570x + 315x2) = 6(8x + -3) + -4(2x + 5) Reorder the terms: -720 + -570x + 315x2 = 6(-3 + 8x) + -4(2x + 5) -720 + -570x + 315x2 = (-3 * 6 + 8x * 6) + -4(2x + 5) -720 + -570x + 315x2 = (-18 + 48x) + -4(2x + 5) Reorder the terms: -720 + -570x + 315x2 = -18 + 48x + -4(5 + 2x) -720 + -570x + 315x2 = -18 + 48x + (5 * -4 + 2x * -4) -720 + -570x + 315x2 = -18 + 48x + (-20 + -8x) Reorder the terms: -720 + -570x + 315x2 = -18 + -20 + 48x + -8x Combine like terms: -18 + -20 = -38 -720 + -570x + 315x2 = -38 + 48x + -8x Combine like terms: 48x + -8x = 40x -720 + -570x + 315x2 = -38 + 40x Solving -720 + -570x + 315x2 = -38 + 40x Solving for variable 'x'. Reorder the terms: -720 + 38 + -570x + -40x + 315x2 = -38 + 40x + 38 + -40x Combine like terms: -720 + 38 = -682 -682 + -570x + -40x + 315x2 = -38 + 40x + 38 + -40x Combine like terms: -570x + -40x = -610x -682 + -610x + 315x2 = -38 + 40x + 38 + -40x Reorder the terms: -682 + -610x + 315x2 = -38 + 38 + 40x + -40x Combine like terms: -38 + 38 = 0 -682 + -610x + 315x2 = 0 + 40x + -40x -682 + -610x + 315x2 = 40x + -40x Combine like terms: 40x + -40x = 0 -682 + -610x + 315x2 = 0 Begin completing the square. Divide all terms by 315 the coefficient of the squared term: Divide each side by '315'. -2.165079365 + -1.936507937x + x2 = 0 Move the constant term to the right: Add '2.165079365' to each side of the equation. -2.165079365 + -1.936507937x + 2.165079365 + x2 = 0 + 2.165079365 Reorder the terms: -2.165079365 + 2.165079365 + -1.936507937x + x2 = 0 + 2.165079365 Combine like terms: -2.165079365 + 2.165079365 = 0.000000000 0.000000000 + -1.936507937x + x2 = 0 + 2.165079365 -1.936507937x + x2 = 0 + 2.165079365 Combine like terms: 0 + 2.165079365 = 2.165079365 -1.936507937x + x2 = 2.165079365 The x term is -1.936507937x. Take half its coefficient (-0.9682539685). Square it (0.9375157475) and add it to both sides. Add '0.9375157475' to each side of the equation. -1.936507937x + 0.9375157475 + x2 = 2.165079365 + 0.9375157475 Reorder the terms: 0.9375157475 + -1.936507937x + x2 = 2.165079365 + 0.9375157475 Combine like terms: 2.165079365 + 0.9375157475 = 3.1025951125 0.9375157475 + -1.936507937x + x2 = 3.1025951125 Factor a perfect square on the left side: (x + -0.9682539685)(x + -0.9682539685) = 3.1025951125 Calculate the square root of the right side: 1.761418494 Break this problem into two subproblems by setting (x + -0.9682539685) equal to 1.761418494 and -1.761418494.Subproblem 1
x + -0.9682539685 = 1.761418494 Simplifying x + -0.9682539685 = 1.761418494 Reorder the terms: -0.9682539685 + x = 1.761418494 Solving -0.9682539685 + x = 1.761418494 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9682539685' to each side of the equation. -0.9682539685 + 0.9682539685 + x = 1.761418494 + 0.9682539685 Combine like terms: -0.9682539685 + 0.9682539685 = 0.0000000000 0.0000000000 + x = 1.761418494 + 0.9682539685 x = 1.761418494 + 0.9682539685 Combine like terms: 1.761418494 + 0.9682539685 = 2.7296724625 x = 2.7296724625 Simplifying x = 2.7296724625Subproblem 2
x + -0.9682539685 = -1.761418494 Simplifying x + -0.9682539685 = -1.761418494 Reorder the terms: -0.9682539685 + x = -1.761418494 Solving -0.9682539685 + x = -1.761418494 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9682539685' to each side of the equation. -0.9682539685 + 0.9682539685 + x = -1.761418494 + 0.9682539685 Combine like terms: -0.9682539685 + 0.9682539685 = 0.0000000000 0.0000000000 + x = -1.761418494 + 0.9682539685 x = -1.761418494 + 0.9682539685 Combine like terms: -1.761418494 + 0.9682539685 = -0.7931645255 x = -0.7931645255 Simplifying x = -0.7931645255Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.7296724625, -0.7931645255}
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