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Simplifying (12 + 2x)(5 + 2x) = 110 Multiply (12 + 2x) * (5 + 2x) (12(5 + 2x) + 2x * (5 + 2x)) = 110 ((5 * 12 + 2x * 12) + 2x * (5 + 2x)) = 110 ((60 + 24x) + 2x * (5 + 2x)) = 110 (60 + 24x + (5 * 2x + 2x * 2x)) = 110 (60 + 24x + (10x + 4x2)) = 110 Combine like terms: 24x + 10x = 34x (60 + 34x + 4x2) = 110 Solving 60 + 34x + 4x2 = 110 Solving for variable 'x'. Reorder the terms: 60 + -110 + 34x + 4x2 = 110 + -110 Combine like terms: 60 + -110 = -50 -50 + 34x + 4x2 = 110 + -110 Combine like terms: 110 + -110 = 0 -50 + 34x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-25 + 17x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-25 + 17x + 2x2)' equal to zero and attempt to solve: Simplifying -25 + 17x + 2x2 = 0 Solving -25 + 17x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12.5 + 8.5x + x2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 8.5x + 12.5 + x2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 8.5x + x2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 8.5x + x2 = 0 + 12.5 8.5x + x2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 8.5x + x2 = 12.5 The x term is 8.5x. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5x + 18.0625 + x2 = 12.5 + 18.0625 Reorder the terms: 18.0625 + 8.5x + x2 = 12.5 + 18.0625 Combine like terms: 12.5 + 18.0625 = 30.5625 18.0625 + 8.5x + x2 = 30.5625 Factor a perfect square on the left side: (x + 4.25)(x + 4.25) = 30.5625 Calculate the square root of the right side: 5.528336097 Break this problem into two subproblems by setting (x + 4.25) equal to 5.528336097 and -5.528336097.Subproblem 1
x + 4.25 = 5.528336097 Simplifying x + 4.25 = 5.528336097 Reorder the terms: 4.25 + x = 5.528336097 Solving 4.25 + x = 5.528336097 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 5.528336097 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 5.528336097 + -4.25 x = 5.528336097 + -4.25 Combine like terms: 5.528336097 + -4.25 = 1.278336097 x = 1.278336097 Simplifying x = 1.278336097Subproblem 2
x + 4.25 = -5.528336097 Simplifying x + 4.25 = -5.528336097 Reorder the terms: 4.25 + x = -5.528336097 Solving 4.25 + x = -5.528336097 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -5.528336097 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -5.528336097 + -4.25 x = -5.528336097 + -4.25 Combine like terms: -5.528336097 + -4.25 = -9.778336097 x = -9.778336097 Simplifying x = -9.778336097Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.278336097, -9.778336097}Solution
x = {1.278336097, -9.778336097}
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