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Simplifying 224 = (2x + 8)(2x + 12) Reorder the terms: 224 = (8 + 2x)(2x + 12) Reorder the terms: 224 = (8 + 2x)(12 + 2x) Multiply (8 + 2x) * (12 + 2x) 224 = (8(12 + 2x) + 2x * (12 + 2x)) 224 = ((12 * 8 + 2x * 8) + 2x * (12 + 2x)) 224 = ((96 + 16x) + 2x * (12 + 2x)) 224 = (96 + 16x + (12 * 2x + 2x * 2x)) 224 = (96 + 16x + (24x + 4x2)) Combine like terms: 16x + 24x = 40x 224 = (96 + 40x + 4x2) Solving 224 = 96 + 40x + 4x2 Solving for variable 'x'. Combine like terms: 224 + -96 = 128 128 + -40x + -4x2 = 96 + 40x + 4x2 + -96 + -40x + -4x2 Reorder the terms: 128 + -40x + -4x2 = 96 + -96 + 40x + -40x + 4x2 + -4x2 Combine like terms: 96 + -96 = 0 128 + -40x + -4x2 = 0 + 40x + -40x + 4x2 + -4x2 128 + -40x + -4x2 = 40x + -40x + 4x2 + -4x2 Combine like terms: 40x + -40x = 0 128 + -40x + -4x2 = 0 + 4x2 + -4x2 128 + -40x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 128 + -40x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(32 + -10x + -1x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(32 + -10x + -1x2)' equal to zero and attempt to solve: Simplifying 32 + -10x + -1x2 = 0 Solving 32 + -10x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -32 + 10x + x2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + 10x + 32 + x2 = 0 + 32 Reorder the terms: -32 + 32 + 10x + x2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 10x + x2 = 0 + 32 10x + x2 = 0 + 32 Combine like terms: 0 + 32 = 32 10x + x2 = 32 The x term is 10x. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10x + 25 + x2 = 32 + 25 Reorder the terms: 25 + 10x + x2 = 32 + 25 Combine like terms: 32 + 25 = 57 25 + 10x + x2 = 57 Factor a perfect square on the left side: (x + 5)(x + 5) = 57 Calculate the square root of the right side: 7.549834435 Break this problem into two subproblems by setting (x + 5) equal to 7.549834435 and -7.549834435.Subproblem 1
x + 5 = 7.549834435 Simplifying x + 5 = 7.549834435 Reorder the terms: 5 + x = 7.549834435 Solving 5 + x = 7.549834435 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = 7.549834435 + -5 Combine like terms: 5 + -5 = 0 0 + x = 7.549834435 + -5 x = 7.549834435 + -5 Combine like terms: 7.549834435 + -5 = 2.549834435 x = 2.549834435 Simplifying x = 2.549834435Subproblem 2
x + 5 = -7.549834435 Simplifying x + 5 = -7.549834435 Reorder the terms: 5 + x = -7.549834435 Solving 5 + x = -7.549834435 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = -7.549834435 + -5 Combine like terms: 5 + -5 = 0 0 + x = -7.549834435 + -5 x = -7.549834435 + -5 Combine like terms: -7.549834435 + -5 = -12.549834435 x = -12.549834435 Simplifying x = -12.549834435Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.549834435, -12.549834435}Solution
x = {2.549834435, -12.549834435}
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