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Simplifying (8x + 40)(32x + -16) = -24 Reorder the terms: (40 + 8x)(32x + -16) = -24 Reorder the terms: (40 + 8x)(-16 + 32x) = -24 Multiply (40 + 8x) * (-16 + 32x) (40(-16 + 32x) + 8x * (-16 + 32x)) = -24 ((-16 * 40 + 32x * 40) + 8x * (-16 + 32x)) = -24 ((-640 + 1280x) + 8x * (-16 + 32x)) = -24 (-640 + 1280x + (-16 * 8x + 32x * 8x)) = -24 (-640 + 1280x + (-128x + 256x2)) = -24 Combine like terms: 1280x + -128x = 1152x (-640 + 1152x + 256x2) = -24 Solving -640 + 1152x + 256x2 = -24 Solving for variable 'x'. Reorder the terms: -640 + 24 + 1152x + 256x2 = -24 + 24 Combine like terms: -640 + 24 = -616 -616 + 1152x + 256x2 = -24 + 24 Combine like terms: -24 + 24 = 0 -616 + 1152x + 256x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-77 + 144x + 32x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-77 + 144x + 32x2)' equal to zero and attempt to solve: Simplifying -77 + 144x + 32x2 = 0 Solving -77 + 144x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -2.40625 + 4.5x + x2 = 0 Move the constant term to the right: Add '2.40625' to each side of the equation. -2.40625 + 4.5x + 2.40625 + x2 = 0 + 2.40625 Reorder the terms: -2.40625 + 2.40625 + 4.5x + x2 = 0 + 2.40625 Combine like terms: -2.40625 + 2.40625 = 0.00000 0.00000 + 4.5x + x2 = 0 + 2.40625 4.5x + x2 = 0 + 2.40625 Combine like terms: 0 + 2.40625 = 2.40625 4.5x + x2 = 2.40625 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 2.40625 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 2.40625 + 5.0625 Combine like terms: 2.40625 + 5.0625 = 7.46875 5.0625 + 4.5x + x2 = 7.46875 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 7.46875 Calculate the square root of the right side: 2.732901389 Break this problem into two subproblems by setting (x + 2.25) equal to 2.732901389 and -2.732901389.Subproblem 1
x + 2.25 = 2.732901389 Simplifying x + 2.25 = 2.732901389 Reorder the terms: 2.25 + x = 2.732901389 Solving 2.25 + x = 2.732901389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 2.732901389 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.732901389 + -2.25 x = 2.732901389 + -2.25 Combine like terms: 2.732901389 + -2.25 = 0.482901389 x = 0.482901389 Simplifying x = 0.482901389Subproblem 2
x + 2.25 = -2.732901389 Simplifying x + 2.25 = -2.732901389 Reorder the terms: 2.25 + x = -2.732901389 Solving 2.25 + x = -2.732901389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -2.732901389 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.732901389 + -2.25 x = -2.732901389 + -2.25 Combine like terms: -2.732901389 + -2.25 = -4.982901389 x = -4.982901389 Simplifying x = -4.982901389Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.482901389, -4.982901389}Solution
x = {0.482901389, -4.982901389}
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