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Simplifying (2x + -1)(3x + 2) + -1(x + 3)(2x + 1) = 0 Reorder the terms: (-1 + 2x)(3x + 2) + -1(x + 3)(2x + 1) = 0 Reorder the terms: (-1 + 2x)(2 + 3x) + -1(x + 3)(2x + 1) = 0 Multiply (-1 + 2x) * (2 + 3x) (-1(2 + 3x) + 2x * (2 + 3x)) + -1(x + 3)(2x + 1) = 0 ((2 * -1 + 3x * -1) + 2x * (2 + 3x)) + -1(x + 3)(2x + 1) = 0 ((-2 + -3x) + 2x * (2 + 3x)) + -1(x + 3)(2x + 1) = 0 (-2 + -3x + (2 * 2x + 3x * 2x)) + -1(x + 3)(2x + 1) = 0 (-2 + -3x + (4x + 6x2)) + -1(x + 3)(2x + 1) = 0 Combine like terms: -3x + 4x = 1x (-2 + 1x + 6x2) + -1(x + 3)(2x + 1) = 0 Reorder the terms: -2 + 1x + 6x2 + -1(3 + x)(2x + 1) = 0 Reorder the terms: -2 + 1x + 6x2 + -1(3 + x)(1 + 2x) = 0 Multiply (3 + x) * (1 + 2x) -2 + 1x + 6x2 + -1(3(1 + 2x) + x(1 + 2x)) = 0 -2 + 1x + 6x2 + -1((1 * 3 + 2x * 3) + x(1 + 2x)) = 0 -2 + 1x + 6x2 + -1((3 + 6x) + x(1 + 2x)) = 0 -2 + 1x + 6x2 + -1(3 + 6x + (1 * x + 2x * x)) = 0 -2 + 1x + 6x2 + -1(3 + 6x + (1x + 2x2)) = 0 Combine like terms: 6x + 1x = 7x -2 + 1x + 6x2 + -1(3 + 7x + 2x2) = 0 -2 + 1x + 6x2 + (3 * -1 + 7x * -1 + 2x2 * -1) = 0 -2 + 1x + 6x2 + (-3 + -7x + -2x2) = 0 Reorder the terms: -2 + -3 + 1x + -7x + 6x2 + -2x2 = 0 Combine like terms: -2 + -3 = -5 -5 + 1x + -7x + 6x2 + -2x2 = 0 Combine like terms: 1x + -7x = -6x -5 + -6x + 6x2 + -2x2 = 0 Combine like terms: 6x2 + -2x2 = 4x2 -5 + -6x + 4x2 = 0 Solving -5 + -6x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + -1.5x + x2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -1.5x + 1.25 + x2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -1.5x + x2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -1.5x + x2 = 0 + 1.25 -1.5x + x2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -1.5x + x2 = 1.25 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 1.25 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 1.25 + 0.5625 Combine like terms: 1.25 + 0.5625 = 1.8125 0.5625 + -1.5x + x2 = 1.8125 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 1.8125 Calculate the square root of the right side: 1.346291202 Break this problem into two subproblems by setting (x + -0.75) equal to 1.346291202 and -1.346291202.Subproblem 1
x + -0.75 = 1.346291202 Simplifying x + -0.75 = 1.346291202 Reorder the terms: -0.75 + x = 1.346291202 Solving -0.75 + x = 1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.346291202 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.346291202 + 0.75 x = 1.346291202 + 0.75 Combine like terms: 1.346291202 + 0.75 = 2.096291202 x = 2.096291202 Simplifying x = 2.096291202Subproblem 2
x + -0.75 = -1.346291202 Simplifying x + -0.75 = -1.346291202 Reorder the terms: -0.75 + x = -1.346291202 Solving -0.75 + x = -1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.346291202 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.346291202 + 0.75 x = -1.346291202 + 0.75 Combine like terms: -1.346291202 + 0.75 = -0.596291202 x = -0.596291202 Simplifying x = -0.596291202Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.096291202, -0.596291202}
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