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Simplifying (12 + 2x)(16 + 2x) = 204 Multiply (12 + 2x) * (16 + 2x) (12(16 + 2x) + 2x * (16 + 2x)) = 204 ((16 * 12 + 2x * 12) + 2x * (16 + 2x)) = 204 ((192 + 24x) + 2x * (16 + 2x)) = 204 (192 + 24x + (16 * 2x + 2x * 2x)) = 204 (192 + 24x + (32x + 4x2)) = 204 Combine like terms: 24x + 32x = 56x (192 + 56x + 4x2) = 204 Solving 192 + 56x + 4x2 = 204 Solving for variable 'x'. Reorder the terms: 192 + -204 + 56x + 4x2 = 204 + -204 Combine like terms: 192 + -204 = -12 -12 + 56x + 4x2 = 204 + -204 Combine like terms: 204 + -204 = 0 -12 + 56x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + 14x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + 14x + x2)' equal to zero and attempt to solve: Simplifying -3 + 14x + x2 = 0 Solving -3 + 14x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 14x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + 14x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 14x + x2 = 0 + 3 14x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 14x + x2 = 3 The x term is 14x. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14x + 49 + x2 = 3 + 49 Reorder the terms: 49 + 14x + x2 = 3 + 49 Combine like terms: 3 + 49 = 52 49 + 14x + x2 = 52 Factor a perfect square on the left side: (x + 7)(x + 7) = 52 Calculate the square root of the right side: 7.211102551 Break this problem into two subproblems by setting (x + 7) equal to 7.211102551 and -7.211102551.Subproblem 1
x + 7 = 7.211102551 Simplifying x + 7 = 7.211102551 Reorder the terms: 7 + x = 7.211102551 Solving 7 + x = 7.211102551 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = 7.211102551 + -7 Combine like terms: 7 + -7 = 0 0 + x = 7.211102551 + -7 x = 7.211102551 + -7 Combine like terms: 7.211102551 + -7 = 0.211102551 x = 0.211102551 Simplifying x = 0.211102551Subproblem 2
x + 7 = -7.211102551 Simplifying x + 7 = -7.211102551 Reorder the terms: 7 + x = -7.211102551 Solving 7 + x = -7.211102551 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = -7.211102551 + -7 Combine like terms: 7 + -7 = 0 0 + x = -7.211102551 + -7 x = -7.211102551 + -7 Combine like terms: -7.211102551 + -7 = -14.211102551 x = -14.211102551 Simplifying x = -14.211102551Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.211102551, -14.211102551}Solution
x = {0.211102551, -14.211102551}
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