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Simplifying (3x + -2)(4x + 3) = 5x(x + -4) Reorder the terms: (-2 + 3x)(4x + 3) = 5x(x + -4) Reorder the terms: (-2 + 3x)(3 + 4x) = 5x(x + -4) Multiply (-2 + 3x) * (3 + 4x) (-2(3 + 4x) + 3x * (3 + 4x)) = 5x(x + -4) ((3 * -2 + 4x * -2) + 3x * (3 + 4x)) = 5x(x + -4) ((-6 + -8x) + 3x * (3 + 4x)) = 5x(x + -4) (-6 + -8x + (3 * 3x + 4x * 3x)) = 5x(x + -4) (-6 + -8x + (9x + 12x2)) = 5x(x + -4) Combine like terms: -8x + 9x = 1x (-6 + 1x + 12x2) = 5x(x + -4) Reorder the terms: -6 + 1x + 12x2 = 5x(-4 + x) -6 + 1x + 12x2 = (-4 * 5x + x * 5x) -6 + 1x + 12x2 = (-20x + 5x2) Solving -6 + 1x + 12x2 = -20x + 5x2 Solving for variable 'x'. Reorder the terms: -6 + 1x + 20x + 12x2 + -5x2 = -20x + 5x2 + 20x + -5x2 Combine like terms: 1x + 20x = 21x -6 + 21x + 12x2 + -5x2 = -20x + 5x2 + 20x + -5x2 Combine like terms: 12x2 + -5x2 = 7x2 -6 + 21x + 7x2 = -20x + 5x2 + 20x + -5x2 Reorder the terms: -6 + 21x + 7x2 = -20x + 20x + 5x2 + -5x2 Combine like terms: -20x + 20x = 0 -6 + 21x + 7x2 = 0 + 5x2 + -5x2 -6 + 21x + 7x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 -6 + 21x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + 3x + x2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + 3x + 0.8571428571 + x2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + 3x + x2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + 3x + x2 = 0 + 0.8571428571 3x + x2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 3x + x2 = 0.8571428571 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 0.8571428571 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 0.8571428571 + 2.25 Combine like terms: 0.8571428571 + 2.25 = 3.1071428571 2.25 + 3x + x2 = 3.1071428571 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 3.1071428571 Calculate the square root of the right side: 1.762708954 Break this problem into two subproblems by setting (x + 1.5) equal to 1.762708954 and -1.762708954.Subproblem 1
x + 1.5 = 1.762708954 Simplifying x + 1.5 = 1.762708954 Reorder the terms: 1.5 + x = 1.762708954 Solving 1.5 + x = 1.762708954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.762708954 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.762708954 + -1.5 x = 1.762708954 + -1.5 Combine like terms: 1.762708954 + -1.5 = 0.262708954 x = 0.262708954 Simplifying x = 0.262708954Subproblem 2
x + 1.5 = -1.762708954 Simplifying x + 1.5 = -1.762708954 Reorder the terms: 1.5 + x = -1.762708954 Solving 1.5 + x = -1.762708954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.762708954 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.762708954 + -1.5 x = -1.762708954 + -1.5 Combine like terms: -1.762708954 + -1.5 = -3.262708954 x = -3.262708954 Simplifying x = -3.262708954Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.262708954, -3.262708954}
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