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Simplifying 8x + -1(x + 3)(4x + -5) = (5 + 2x)(4 + -1x) + -3(x + 4) Reorder the terms: 8x + -1(3 + x)(4x + -5) = (5 + 2x)(4 + -1x) + -3(x + 4) Reorder the terms: 8x + -1(3 + x)(-5 + 4x) = (5 + 2x)(4 + -1x) + -3(x + 4) Multiply (3 + x) * (-5 + 4x) 8x + -1(3(-5 + 4x) + x(-5 + 4x)) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + -1((-5 * 3 + 4x * 3) + x(-5 + 4x)) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + -1((-15 + 12x) + x(-5 + 4x)) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + -1(-15 + 12x + (-5 * x + 4x * x)) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + -1(-15 + 12x + (-5x + 4x2)) = (5 + 2x)(4 + -1x) + -3(x + 4) Combine like terms: 12x + -5x = 7x 8x + -1(-15 + 7x + 4x2) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + (-15 * -1 + 7x * -1 + 4x2 * -1) = (5 + 2x)(4 + -1x) + -3(x + 4) 8x + (15 + -7x + -4x2) = (5 + 2x)(4 + -1x) + -3(x + 4) Reorder the terms: 15 + 8x + -7x + -4x2 = (5 + 2x)(4 + -1x) + -3(x + 4) Combine like terms: 8x + -7x = 1x 15 + 1x + -4x2 = (5 + 2x)(4 + -1x) + -3(x + 4) Multiply (5 + 2x) * (4 + -1x) 15 + 1x + -4x2 = (5(4 + -1x) + 2x * (4 + -1x)) + -3(x + 4) 15 + 1x + -4x2 = ((4 * 5 + -1x * 5) + 2x * (4 + -1x)) + -3(x + 4) 15 + 1x + -4x2 = ((20 + -5x) + 2x * (4 + -1x)) + -3(x + 4) 15 + 1x + -4x2 = (20 + -5x + (4 * 2x + -1x * 2x)) + -3(x + 4) 15 + 1x + -4x2 = (20 + -5x + (8x + -2x2)) + -3(x + 4) Combine like terms: -5x + 8x = 3x 15 + 1x + -4x2 = (20 + 3x + -2x2) + -3(x + 4) Reorder the terms: 15 + 1x + -4x2 = 20 + 3x + -2x2 + -3(4 + x) 15 + 1x + -4x2 = 20 + 3x + -2x2 + (4 * -3 + x * -3) 15 + 1x + -4x2 = 20 + 3x + -2x2 + (-12 + -3x) Reorder the terms: 15 + 1x + -4x2 = 20 + -12 + 3x + -3x + -2x2 Combine like terms: 20 + -12 = 8 15 + 1x + -4x2 = 8 + 3x + -3x + -2x2 Combine like terms: 3x + -3x = 0 15 + 1x + -4x2 = 8 + 0 + -2x2 15 + 1x + -4x2 = 8 + -2x2 Solving 15 + 1x + -4x2 = 8 + -2x2 Solving for variable 'x'. Reorder the terms: 15 + -8 + 1x + -4x2 + 2x2 = 8 + -2x2 + -8 + 2x2 Combine like terms: 15 + -8 = 7 7 + 1x + -4x2 + 2x2 = 8 + -2x2 + -8 + 2x2 Combine like terms: -4x2 + 2x2 = -2x2 7 + 1x + -2x2 = 8 + -2x2 + -8 + 2x2 Reorder the terms: 7 + 1x + -2x2 = 8 + -8 + -2x2 + 2x2 Combine like terms: 8 + -8 = 0 7 + 1x + -2x2 = 0 + -2x2 + 2x2 7 + 1x + -2x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 7 + 1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3.5 + -0.5x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -0.5x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -0.5x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -0.5x + x2 = 0 + 3.5 -0.5x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -0.5x + x2 = 3.5 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 3.5 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 3.5 + 0.0625 Combine like terms: 3.5 + 0.0625 = 3.5625 0.0625 + -0.5x + x2 = 3.5625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (x + -0.25) equal to 1.887458609 and -1.887458609.Subproblem 1
x + -0.25 = 1.887458609 Simplifying x + -0.25 = 1.887458609 Reorder the terms: -0.25 + x = 1.887458609 Solving -0.25 + x = 1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.887458609 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.887458609 + 0.25 x = 1.887458609 + 0.25 Combine like terms: 1.887458609 + 0.25 = 2.137458609 x = 2.137458609 Simplifying x = 2.137458609Subproblem 2
x + -0.25 = -1.887458609 Simplifying x + -0.25 = -1.887458609 Reorder the terms: -0.25 + x = -1.887458609 Solving -0.25 + x = -1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.887458609 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.887458609 + 0.25 x = -1.887458609 + 0.25 Combine like terms: -1.887458609 + 0.25 = -1.637458609 x = -1.637458609 Simplifying x = -1.637458609Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.137458609, -1.637458609}
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