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Simplifying (3x + 1)(4x + -3) + -1(15x + 5x + 8) = 0 Reorder the terms: (1 + 3x)(4x + -3) + -1(15x + 5x + 8) = 0 Reorder the terms: (1 + 3x)(-3 + 4x) + -1(15x + 5x + 8) = 0 Multiply (1 + 3x) * (-3 + 4x) (1(-3 + 4x) + 3x * (-3 + 4x)) + -1(15x + 5x + 8) = 0 ((-3 * 1 + 4x * 1) + 3x * (-3 + 4x)) + -1(15x + 5x + 8) = 0 ((-3 + 4x) + 3x * (-3 + 4x)) + -1(15x + 5x + 8) = 0 (-3 + 4x + (-3 * 3x + 4x * 3x)) + -1(15x + 5x + 8) = 0 (-3 + 4x + (-9x + 12x2)) + -1(15x + 5x + 8) = 0 Combine like terms: 4x + -9x = -5x (-3 + -5x + 12x2) + -1(15x + 5x + 8) = 0 Reorder the terms: -3 + -5x + 12x2 + -1(8 + 15x + 5x) = 0 Combine like terms: 15x + 5x = 20x -3 + -5x + 12x2 + -1(8 + 20x) = 0 -3 + -5x + 12x2 + (8 * -1 + 20x * -1) = 0 -3 + -5x + 12x2 + (-8 + -20x) = 0 Reorder the terms: -3 + -8 + -5x + -20x + 12x2 = 0 Combine like terms: -3 + -8 = -11 -11 + -5x + -20x + 12x2 = 0 Combine like terms: -5x + -20x = -25x -11 + -25x + 12x2 = 0 Solving -11 + -25x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.9166666667 + -2.083333333x + x2 = 0 Move the constant term to the right: Add '0.9166666667' to each side of the equation. -0.9166666667 + -2.083333333x + 0.9166666667 + x2 = 0 + 0.9166666667 Reorder the terms: -0.9166666667 + 0.9166666667 + -2.083333333x + x2 = 0 + 0.9166666667 Combine like terms: -0.9166666667 + 0.9166666667 = 0.0000000000 0.0000000000 + -2.083333333x + x2 = 0 + 0.9166666667 -2.083333333x + x2 = 0 + 0.9166666667 Combine like terms: 0 + 0.9166666667 = 0.9166666667 -2.083333333x + x2 = 0.9166666667 The x term is -2.083333333x. Take half its coefficient (-1.041666667). Square it (1.085069445) and add it to both sides. Add '1.085069445' to each side of the equation. -2.083333333x + 1.085069445 + x2 = 0.9166666667 + 1.085069445 Reorder the terms: 1.085069445 + -2.083333333x + x2 = 0.9166666667 + 1.085069445 Combine like terms: 0.9166666667 + 1.085069445 = 2.0017361117 1.085069445 + -2.083333333x + x2 = 2.0017361117 Factor a perfect square on the left side: (x + -1.041666667)(x + -1.041666667) = 2.0017361117 Calculate the square root of the right side: 1.414827237 Break this problem into two subproblems by setting (x + -1.041666667) equal to 1.414827237 and -1.414827237.Subproblem 1
x + -1.041666667 = 1.414827237 Simplifying x + -1.041666667 = 1.414827237 Reorder the terms: -1.041666667 + x = 1.414827237 Solving -1.041666667 + x = 1.414827237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.041666667' to each side of the equation. -1.041666667 + 1.041666667 + x = 1.414827237 + 1.041666667 Combine like terms: -1.041666667 + 1.041666667 = 0.000000000 0.000000000 + x = 1.414827237 + 1.041666667 x = 1.414827237 + 1.041666667 Combine like terms: 1.414827237 + 1.041666667 = 2.456493904 x = 2.456493904 Simplifying x = 2.456493904Subproblem 2
x + -1.041666667 = -1.414827237 Simplifying x + -1.041666667 = -1.414827237 Reorder the terms: -1.041666667 + x = -1.414827237 Solving -1.041666667 + x = -1.414827237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.041666667' to each side of the equation. -1.041666667 + 1.041666667 + x = -1.414827237 + 1.041666667 Combine like terms: -1.041666667 + 1.041666667 = 0.000000000 0.000000000 + x = -1.414827237 + 1.041666667 x = -1.414827237 + 1.041666667 Combine like terms: -1.414827237 + 1.041666667 = -0.37316057 x = -0.37316057 Simplifying x = -0.37316057Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.456493904, -0.37316057}
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