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Simplifying (2 + -3x) * 2 + (1 + 4x) * 2 = (5x + -1)(5x + -1) Reorder the terms for easier multiplication: 2(2 + -3x) + (1 + 4x) * 2 = (5x + -1)(5x + -1) (2 * 2 + -3x * 2) + (1 + 4x) * 2 = (5x + -1)(5x + -1) (4 + -6x) + (1 + 4x) * 2 = (5x + -1)(5x + -1) Reorder the terms for easier multiplication: 4 + -6x + 2(1 + 4x) = (5x + -1)(5x + -1) 4 + -6x + (1 * 2 + 4x * 2) = (5x + -1)(5x + -1) 4 + -6x + (2 + 8x) = (5x + -1)(5x + -1) Reorder the terms: 4 + 2 + -6x + 8x = (5x + -1)(5x + -1) Combine like terms: 4 + 2 = 6 6 + -6x + 8x = (5x + -1)(5x + -1) Combine like terms: -6x + 8x = 2x 6 + 2x = (5x + -1)(5x + -1) Reorder the terms: 6 + 2x = (-1 + 5x)(5x + -1) Reorder the terms: 6 + 2x = (-1 + 5x)(-1 + 5x) Multiply (-1 + 5x) * (-1 + 5x) 6 + 2x = (-1(-1 + 5x) + 5x * (-1 + 5x)) 6 + 2x = ((-1 * -1 + 5x * -1) + 5x * (-1 + 5x)) 6 + 2x = ((1 + -5x) + 5x * (-1 + 5x)) 6 + 2x = (1 + -5x + (-1 * 5x + 5x * 5x)) 6 + 2x = (1 + -5x + (-5x + 25x2)) Combine like terms: -5x + -5x = -10x 6 + 2x = (1 + -10x + 25x2) Solving 6 + 2x = 1 + -10x + 25x2 Solving for variable 'x'. Reorder the terms: 6 + -1 + 2x + 10x + -25x2 = 1 + -10x + 25x2 + -1 + 10x + -25x2 Combine like terms: 6 + -1 = 5 5 + 2x + 10x + -25x2 = 1 + -10x + 25x2 + -1 + 10x + -25x2 Combine like terms: 2x + 10x = 12x 5 + 12x + -25x2 = 1 + -10x + 25x2 + -1 + 10x + -25x2 Reorder the terms: 5 + 12x + -25x2 = 1 + -1 + -10x + 10x + 25x2 + -25x2 Combine like terms: 1 + -1 = 0 5 + 12x + -25x2 = 0 + -10x + 10x + 25x2 + -25x2 5 + 12x + -25x2 = -10x + 10x + 25x2 + -25x2 Combine like terms: -10x + 10x = 0 5 + 12x + -25x2 = 0 + 25x2 + -25x2 5 + 12x + -25x2 = 25x2 + -25x2 Combine like terms: 25x2 + -25x2 = 0 5 + 12x + -25x2 = 0 Begin completing the square. Divide all terms by -25 the coefficient of the squared term: Divide each side by '-25'. -0.2 + -0.48x + x2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.48x + 0.2 + x2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.48x + x2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.48x + x2 = 0 + 0.2 -0.48x + x2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -0.48x + x2 = 0.2 The x term is -0.48x. Take half its coefficient (-0.24). Square it (0.0576) and add it to both sides. Add '0.0576' to each side of the equation. -0.48x + 0.0576 + x2 = 0.2 + 0.0576 Reorder the terms: 0.0576 + -0.48x + x2 = 0.2 + 0.0576 Combine like terms: 0.2 + 0.0576 = 0.2576 0.0576 + -0.48x + x2 = 0.2576 Factor a perfect square on the left side: (x + -0.24)(x + -0.24) = 0.2576 Calculate the square root of the right side: 0.507543102 Break this problem into two subproblems by setting (x + -0.24) equal to 0.507543102 and -0.507543102.Subproblem 1
x + -0.24 = 0.507543102 Simplifying x + -0.24 = 0.507543102 Reorder the terms: -0.24 + x = 0.507543102 Solving -0.24 + x = 0.507543102 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.24' to each side of the equation. -0.24 + 0.24 + x = 0.507543102 + 0.24 Combine like terms: -0.24 + 0.24 = 0.00 0.00 + x = 0.507543102 + 0.24 x = 0.507543102 + 0.24 Combine like terms: 0.507543102 + 0.24 = 0.747543102 x = 0.747543102 Simplifying x = 0.747543102Subproblem 2
x + -0.24 = -0.507543102 Simplifying x + -0.24 = -0.507543102 Reorder the terms: -0.24 + x = -0.507543102 Solving -0.24 + x = -0.507543102 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.24' to each side of the equation. -0.24 + 0.24 + x = -0.507543102 + 0.24 Combine like terms: -0.24 + 0.24 = 0.00 0.00 + x = -0.507543102 + 0.24 x = -0.507543102 + 0.24 Combine like terms: -0.507543102 + 0.24 = -0.267543102 x = -0.267543102 Simplifying x = -0.267543102Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.747543102, -0.267543102}
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