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Simplifying (-3x + 1)(-1x + 4) + 5x2 + -6 = 0 Reorder the terms: (1 + -3x)(-1x + 4) + 5x2 + -6 = 0 Reorder the terms: (1 + -3x)(4 + -1x) + 5x2 + -6 = 0 Multiply (1 + -3x) * (4 + -1x) (1(4 + -1x) + -3x * (4 + -1x)) + 5x2 + -6 = 0 ((4 * 1 + -1x * 1) + -3x * (4 + -1x)) + 5x2 + -6 = 0 ((4 + -1x) + -3x * (4 + -1x)) + 5x2 + -6 = 0 (4 + -1x + (4 * -3x + -1x * -3x)) + 5x2 + -6 = 0 (4 + -1x + (-12x + 3x2)) + 5x2 + -6 = 0 Combine like terms: -1x + -12x = -13x (4 + -13x + 3x2) + 5x2 + -6 = 0 Reorder the terms: 4 + -6 + -13x + 3x2 + 5x2 = 0 Combine like terms: 4 + -6 = -2 -2 + -13x + 3x2 + 5x2 = 0 Combine like terms: 3x2 + 5x2 = 8x2 -2 + -13x + 8x2 = 0 Solving -2 + -13x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.25 + -1.625x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.625x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.625x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.625x + x2 = 0 + 0.25 -1.625x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.625x + x2 = 0.25 The x term is -1.625x. Take half its coefficient (-0.8125). Square it (0.66015625) and add it to both sides. Add '0.66015625' to each side of the equation. -1.625x + 0.66015625 + x2 = 0.25 + 0.66015625 Reorder the terms: 0.66015625 + -1.625x + x2 = 0.25 + 0.66015625 Combine like terms: 0.25 + 0.66015625 = 0.91015625 0.66015625 + -1.625x + x2 = 0.91015625 Factor a perfect square on the left side: (x + -0.8125)(x + -0.8125) = 0.91015625 Calculate the square root of the right side: 0.954021095 Break this problem into two subproblems by setting (x + -0.8125) equal to 0.954021095 and -0.954021095.Subproblem 1
x + -0.8125 = 0.954021095 Simplifying x + -0.8125 = 0.954021095 Reorder the terms: -0.8125 + x = 0.954021095 Solving -0.8125 + x = 0.954021095 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8125' to each side of the equation. -0.8125 + 0.8125 + x = 0.954021095 + 0.8125 Combine like terms: -0.8125 + 0.8125 = 0.0000 0.0000 + x = 0.954021095 + 0.8125 x = 0.954021095 + 0.8125 Combine like terms: 0.954021095 + 0.8125 = 1.766521095 x = 1.766521095 Simplifying x = 1.766521095Subproblem 2
x + -0.8125 = -0.954021095 Simplifying x + -0.8125 = -0.954021095 Reorder the terms: -0.8125 + x = -0.954021095 Solving -0.8125 + x = -0.954021095 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8125' to each side of the equation. -0.8125 + 0.8125 + x = -0.954021095 + 0.8125 Combine like terms: -0.8125 + 0.8125 = 0.0000 0.0000 + x = -0.954021095 + 0.8125 x = -0.954021095 + 0.8125 Combine like terms: -0.954021095 + 0.8125 = -0.141521095 x = -0.141521095 Simplifying x = -0.141521095Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.766521095, -0.141521095}
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