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Simplifying 110 = (x + 5)(2x + 12) Reorder the terms: 110 = (5 + x)(2x + 12) Reorder the terms: 110 = (5 + x)(12 + 2x) Multiply (5 + x) * (12 + 2x) 110 = (5(12 + 2x) + x(12 + 2x)) 110 = ((12 * 5 + 2x * 5) + x(12 + 2x)) 110 = ((60 + 10x) + x(12 + 2x)) 110 = (60 + 10x + (12 * x + 2x * x)) 110 = (60 + 10x + (12x + 2x2)) Combine like terms: 10x + 12x = 22x 110 = (60 + 22x + 2x2) Solving 110 = 60 + 22x + 2x2 Solving for variable 'x'. Combine like terms: 110 + -60 = 50 50 + -22x + -2x2 = 60 + 22x + 2x2 + -60 + -22x + -2x2 Reorder the terms: 50 + -22x + -2x2 = 60 + -60 + 22x + -22x + 2x2 + -2x2 Combine like terms: 60 + -60 = 0 50 + -22x + -2x2 = 0 + 22x + -22x + 2x2 + -2x2 50 + -22x + -2x2 = 22x + -22x + 2x2 + -2x2 Combine like terms: 22x + -22x = 0 50 + -22x + -2x2 = 0 + 2x2 + -2x2 50 + -22x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 50 + -22x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(25 + -11x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(25 + -11x + -1x2)' equal to zero and attempt to solve: Simplifying 25 + -11x + -1x2 = 0 Solving 25 + -11x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -25 + 11x + x2 = 0 Move the constant term to the right: Add '25' to each side of the equation. -25 + 11x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + 11x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 11x + x2 = 0 + 25 11x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 11x + x2 = 25 The x term is 11x. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11x + 30.25 + x2 = 25 + 30.25 Reorder the terms: 30.25 + 11x + x2 = 25 + 30.25 Combine like terms: 25 + 30.25 = 55.25 30.25 + 11x + x2 = 55.25 Factor a perfect square on the left side: (x + 5.5)(x + 5.5) = 55.25 Calculate the square root of the right side: 7.433034374 Break this problem into two subproblems by setting (x + 5.5) equal to 7.433034374 and -7.433034374.Subproblem 1
x + 5.5 = 7.433034374 Simplifying x + 5.5 = 7.433034374 Reorder the terms: 5.5 + x = 7.433034374 Solving 5.5 + x = 7.433034374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = 7.433034374 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = 7.433034374 + -5.5 x = 7.433034374 + -5.5 Combine like terms: 7.433034374 + -5.5 = 1.933034374 x = 1.933034374 Simplifying x = 1.933034374Subproblem 2
x + 5.5 = -7.433034374 Simplifying x + 5.5 = -7.433034374 Reorder the terms: 5.5 + x = -7.433034374 Solving 5.5 + x = -7.433034374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = -7.433034374 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = -7.433034374 + -5.5 x = -7.433034374 + -5.5 Combine like terms: -7.433034374 + -5.5 = -12.933034374 x = -12.933034374 Simplifying x = -12.933034374Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.933034374, -12.933034374}Solution
x = {1.933034374, -12.933034374}
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