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Simplifying 6(x + 2)(x + -4) + -5(x + 3)(x + -4) = 6(x + 3) Reorder the terms: 6(2 + x)(x + -4) + -5(x + 3)(x + -4) = 6(x + 3) Reorder the terms: 6(2 + x)(-4 + x) + -5(x + 3)(x + -4) = 6(x + 3) Multiply (2 + x) * (-4 + x) 6(2(-4 + x) + x(-4 + x)) + -5(x + 3)(x + -4) = 6(x + 3) 6((-4 * 2 + x * 2) + x(-4 + x)) + -5(x + 3)(x + -4) = 6(x + 3) 6((-8 + 2x) + x(-4 + x)) + -5(x + 3)(x + -4) = 6(x + 3) 6(-8 + 2x + (-4 * x + x * x)) + -5(x + 3)(x + -4) = 6(x + 3) 6(-8 + 2x + (-4x + x2)) + -5(x + 3)(x + -4) = 6(x + 3) Combine like terms: 2x + -4x = -2x 6(-8 + -2x + x2) + -5(x + 3)(x + -4) = 6(x + 3) (-8 * 6 + -2x * 6 + x2 * 6) + -5(x + 3)(x + -4) = 6(x + 3) (-48 + -12x + 6x2) + -5(x + 3)(x + -4) = 6(x + 3) Reorder the terms: -48 + -12x + 6x2 + -5(3 + x)(x + -4) = 6(x + 3) Reorder the terms: -48 + -12x + 6x2 + -5(3 + x)(-4 + x) = 6(x + 3) Multiply (3 + x) * (-4 + x) -48 + -12x + 6x2 + -5(3(-4 + x) + x(-4 + x)) = 6(x + 3) -48 + -12x + 6x2 + -5((-4 * 3 + x * 3) + x(-4 + x)) = 6(x + 3) -48 + -12x + 6x2 + -5((-12 + 3x) + x(-4 + x)) = 6(x + 3) -48 + -12x + 6x2 + -5(-12 + 3x + (-4 * x + x * x)) = 6(x + 3) -48 + -12x + 6x2 + -5(-12 + 3x + (-4x + x2)) = 6(x + 3) Combine like terms: 3x + -4x = -1x -48 + -12x + 6x2 + -5(-12 + -1x + x2) = 6(x + 3) -48 + -12x + 6x2 + (-12 * -5 + -1x * -5 + x2 * -5) = 6(x + 3) -48 + -12x + 6x2 + (60 + 5x + -5x2) = 6(x + 3) Reorder the terms: -48 + 60 + -12x + 5x + 6x2 + -5x2 = 6(x + 3) Combine like terms: -48 + 60 = 12 12 + -12x + 5x + 6x2 + -5x2 = 6(x + 3) Combine like terms: -12x + 5x = -7x 12 + -7x + 6x2 + -5x2 = 6(x + 3) Combine like terms: 6x2 + -5x2 = 1x2 12 + -7x + 1x2 = 6(x + 3) Reorder the terms: 12 + -7x + 1x2 = 6(3 + x) 12 + -7x + 1x2 = (3 * 6 + x * 6) 12 + -7x + 1x2 = (18 + 6x) Solving 12 + -7x + 1x2 = 18 + 6x Solving for variable 'x'. Reorder the terms: 12 + -18 + -7x + -6x + 1x2 = 18 + 6x + -18 + -6x Combine like terms: 12 + -18 = -6 -6 + -7x + -6x + 1x2 = 18 + 6x + -18 + -6x Combine like terms: -7x + -6x = -13x -6 + -13x + 1x2 = 18 + 6x + -18 + -6x Reorder the terms: -6 + -13x + 1x2 = 18 + -18 + 6x + -6x Combine like terms: 18 + -18 = 0 -6 + -13x + 1x2 = 0 + 6x + -6x -6 + -13x + 1x2 = 6x + -6x Combine like terms: 6x + -6x = 0 -6 + -13x + 1x2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -13x + 6 + x2 = 0 + 6 Reorder the terms: -6 + 6 + -13x + x2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -13x + x2 = 0 + 6 -13x + x2 = 0 + 6 Combine like terms: 0 + 6 = 6 -13x + x2 = 6 The x term is -13x. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13x + 42.25 + x2 = 6 + 42.25 Reorder the terms: 42.25 + -13x + x2 = 6 + 42.25 Combine like terms: 6 + 42.25 = 48.25 42.25 + -13x + x2 = 48.25 Factor a perfect square on the left side: (x + -6.5)(x + -6.5) = 48.25 Calculate the square root of the right side: 6.946221995 Break this problem into two subproblems by setting (x + -6.5) equal to 6.946221995 and -6.946221995.Subproblem 1
x + -6.5 = 6.946221995 Simplifying x + -6.5 = 6.946221995 Reorder the terms: -6.5 + x = 6.946221995 Solving -6.5 + x = 6.946221995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = 6.946221995 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = 6.946221995 + 6.5 x = 6.946221995 + 6.5 Combine like terms: 6.946221995 + 6.5 = 13.446221995 x = 13.446221995 Simplifying x = 13.446221995Subproblem 2
x + -6.5 = -6.946221995 Simplifying x + -6.5 = -6.946221995 Reorder the terms: -6.5 + x = -6.946221995 Solving -6.5 + x = -6.946221995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = -6.946221995 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = -6.946221995 + 6.5 x = -6.946221995 + 6.5 Combine like terms: -6.946221995 + 6.5 = -0.446221995 x = -0.446221995 Simplifying x = -0.446221995Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.446221995, -0.446221995}
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