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Simplifying 1(2x + -1) + 1(3x + 1) = (6x + 2)(2x + -1) Reorder the terms: 1(-1 + 2x) + 1(3x + 1) = (6x + 2)(2x + -1) (-1 * 1 + 2x * 1) + 1(3x + 1) = (6x + 2)(2x + -1) (-1 + 2x) + 1(3x + 1) = (6x + 2)(2x + -1) Reorder the terms: -1 + 2x + 1(1 + 3x) = (6x + 2)(2x + -1) -1 + 2x + (1 * 1 + 3x * 1) = (6x + 2)(2x + -1) -1 + 2x + (1 + 3x) = (6x + 2)(2x + -1) Reorder the terms: -1 + 1 + 2x + 3x = (6x + 2)(2x + -1) Combine like terms: -1 + 1 = 0 0 + 2x + 3x = (6x + 2)(2x + -1) 2x + 3x = (6x + 2)(2x + -1) Combine like terms: 2x + 3x = 5x 5x = (6x + 2)(2x + -1) Reorder the terms: 5x = (2 + 6x)(2x + -1) Reorder the terms: 5x = (2 + 6x)(-1 + 2x) Multiply (2 + 6x) * (-1 + 2x) 5x = (2(-1 + 2x) + 6x * (-1 + 2x)) 5x = ((-1 * 2 + 2x * 2) + 6x * (-1 + 2x)) 5x = ((-2 + 4x) + 6x * (-1 + 2x)) 5x = (-2 + 4x + (-1 * 6x + 2x * 6x)) 5x = (-2 + 4x + (-6x + 12x2)) Combine like terms: 4x + -6x = -2x 5x = (-2 + -2x + 12x2) Solving 5x = -2 + -2x + 12x2 Solving for variable 'x'. Reorder the terms: 2 + 5x + 2x + -12x2 = -2 + -2x + 12x2 + 2 + 2x + -12x2 Combine like terms: 5x + 2x = 7x 2 + 7x + -12x2 = -2 + -2x + 12x2 + 2 + 2x + -12x2 Reorder the terms: 2 + 7x + -12x2 = -2 + 2 + -2x + 2x + 12x2 + -12x2 Combine like terms: -2 + 2 = 0 2 + 7x + -12x2 = 0 + -2x + 2x + 12x2 + -12x2 2 + 7x + -12x2 = -2x + 2x + 12x2 + -12x2 Combine like terms: -2x + 2x = 0 2 + 7x + -12x2 = 0 + 12x2 + -12x2 2 + 7x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 2 + 7x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -0.1666666667 + -0.5833333333x + x2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -0.5833333333x + 0.1666666667 + x2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -0.5833333333x + x2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -0.5833333333x + x2 = 0 + 0.1666666667 -0.5833333333x + x2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -0.5833333333x + x2 = 0.1666666667 The x term is -0.5833333333x. Take half its coefficient (-0.2916666667). Square it (0.08506944446) and add it to both sides. Add '0.08506944446' to each side of the equation. -0.5833333333x + 0.08506944446 + x2 = 0.1666666667 + 0.08506944446 Reorder the terms: 0.08506944446 + -0.5833333333x + x2 = 0.1666666667 + 0.08506944446 Combine like terms: 0.1666666667 + 0.08506944446 = 0.25173611116 0.08506944446 + -0.5833333333x + x2 = 0.25173611116 Factor a perfect square on the left side: (x + -0.2916666667)(x + -0.2916666667) = 0.25173611116 Calculate the square root of the right side: 0.501733107 Break this problem into two subproblems by setting (x + -0.2916666667) equal to 0.501733107 and -0.501733107.Subproblem 1
x + -0.2916666667 = 0.501733107 Simplifying x + -0.2916666667 = 0.501733107 Reorder the terms: -0.2916666667 + x = 0.501733107 Solving -0.2916666667 + x = 0.501733107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + x = 0.501733107 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + x = 0.501733107 + 0.2916666667 x = 0.501733107 + 0.2916666667 Combine like terms: 0.501733107 + 0.2916666667 = 0.7933997737 x = 0.7933997737 Simplifying x = 0.7933997737Subproblem 2
x + -0.2916666667 = -0.501733107 Simplifying x + -0.2916666667 = -0.501733107 Reorder the terms: -0.2916666667 + x = -0.501733107 Solving -0.2916666667 + x = -0.501733107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + x = -0.501733107 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + x = -0.501733107 + 0.2916666667 x = -0.501733107 + 0.2916666667 Combine like terms: -0.501733107 + 0.2916666667 = -0.2100664403 x = -0.2100664403 Simplifying x = -0.2100664403Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7933997737, -0.2100664403}
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