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Simplifying (5 + -3x)(-4x + 6) = (8x + 11)(3x + -6) Reorder the terms: (5 + -3x)(6 + -4x) = (8x + 11)(3x + -6) Multiply (5 + -3x) * (6 + -4x) (5(6 + -4x) + -3x * (6 + -4x)) = (8x + 11)(3x + -6) ((6 * 5 + -4x * 5) + -3x * (6 + -4x)) = (8x + 11)(3x + -6) ((30 + -20x) + -3x * (6 + -4x)) = (8x + 11)(3x + -6) (30 + -20x + (6 * -3x + -4x * -3x)) = (8x + 11)(3x + -6) (30 + -20x + (-18x + 12x2)) = (8x + 11)(3x + -6) Combine like terms: -20x + -18x = -38x (30 + -38x + 12x2) = (8x + 11)(3x + -6) Reorder the terms: 30 + -38x + 12x2 = (11 + 8x)(3x + -6) Reorder the terms: 30 + -38x + 12x2 = (11 + 8x)(-6 + 3x) Multiply (11 + 8x) * (-6 + 3x) 30 + -38x + 12x2 = (11(-6 + 3x) + 8x * (-6 + 3x)) 30 + -38x + 12x2 = ((-6 * 11 + 3x * 11) + 8x * (-6 + 3x)) 30 + -38x + 12x2 = ((-66 + 33x) + 8x * (-6 + 3x)) 30 + -38x + 12x2 = (-66 + 33x + (-6 * 8x + 3x * 8x)) 30 + -38x + 12x2 = (-66 + 33x + (-48x + 24x2)) Combine like terms: 33x + -48x = -15x 30 + -38x + 12x2 = (-66 + -15x + 24x2) Solving 30 + -38x + 12x2 = -66 + -15x + 24x2 Solving for variable 'x'. Reorder the terms: 30 + 66 + -38x + 15x + 12x2 + -24x2 = -66 + -15x + 24x2 + 66 + 15x + -24x2 Combine like terms: 30 + 66 = 96 96 + -38x + 15x + 12x2 + -24x2 = -66 + -15x + 24x2 + 66 + 15x + -24x2 Combine like terms: -38x + 15x = -23x 96 + -23x + 12x2 + -24x2 = -66 + -15x + 24x2 + 66 + 15x + -24x2 Combine like terms: 12x2 + -24x2 = -12x2 96 + -23x + -12x2 = -66 + -15x + 24x2 + 66 + 15x + -24x2 Reorder the terms: 96 + -23x + -12x2 = -66 + 66 + -15x + 15x + 24x2 + -24x2 Combine like terms: -66 + 66 = 0 96 + -23x + -12x2 = 0 + -15x + 15x + 24x2 + -24x2 96 + -23x + -12x2 = -15x + 15x + 24x2 + -24x2 Combine like terms: -15x + 15x = 0 96 + -23x + -12x2 = 0 + 24x2 + -24x2 96 + -23x + -12x2 = 24x2 + -24x2 Combine like terms: 24x2 + -24x2 = 0 96 + -23x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -8 + 1.916666667x + x2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 1.916666667x + 8 + x2 = 0 + 8 Reorder the terms: -8 + 8 + 1.916666667x + x2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 1.916666667x + x2 = 0 + 8 1.916666667x + x2 = 0 + 8 Combine like terms: 0 + 8 = 8 1.916666667x + x2 = 8 The x term is 1.916666667x. Take half its coefficient (0.9583333335). Square it (0.9184027781) and add it to both sides. Add '0.9184027781' to each side of the equation. 1.916666667x + 0.9184027781 + x2 = 8 + 0.9184027781 Reorder the terms: 0.9184027781 + 1.916666667x + x2 = 8 + 0.9184027781 Combine like terms: 8 + 0.9184027781 = 8.9184027781 0.9184027781 + 1.916666667x + x2 = 8.9184027781 Factor a perfect square on the left side: (x + 0.9583333335)(x + 0.9583333335) = 8.9184027781 Calculate the square root of the right side: 2.986369498 Break this problem into two subproblems by setting (x + 0.9583333335) equal to 2.986369498 and -2.986369498.Subproblem 1
x + 0.9583333335 = 2.986369498 Simplifying x + 0.9583333335 = 2.986369498 Reorder the terms: 0.9583333335 + x = 2.986369498 Solving 0.9583333335 + x = 2.986369498 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9583333335' to each side of the equation. 0.9583333335 + -0.9583333335 + x = 2.986369498 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + x = 2.986369498 + -0.9583333335 x = 2.986369498 + -0.9583333335 Combine like terms: 2.986369498 + -0.9583333335 = 2.0280361645 x = 2.0280361645 Simplifying x = 2.0280361645Subproblem 2
x + 0.9583333335 = -2.986369498 Simplifying x + 0.9583333335 = -2.986369498 Reorder the terms: 0.9583333335 + x = -2.986369498 Solving 0.9583333335 + x = -2.986369498 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9583333335' to each side of the equation. 0.9583333335 + -0.9583333335 + x = -2.986369498 + -0.9583333335 Combine like terms: 0.9583333335 + -0.9583333335 = 0.0000000000 0.0000000000 + x = -2.986369498 + -0.9583333335 x = -2.986369498 + -0.9583333335 Combine like terms: -2.986369498 + -0.9583333335 = -3.9447028315 x = -3.9447028315 Simplifying x = -3.9447028315Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.0280361645, -3.9447028315}
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