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Simplifying (12 + 2x)(9 + 2x) = 128 Multiply (12 + 2x) * (9 + 2x) (12(9 + 2x) + 2x * (9 + 2x)) = 128 ((9 * 12 + 2x * 12) + 2x * (9 + 2x)) = 128 ((108 + 24x) + 2x * (9 + 2x)) = 128 (108 + 24x + (9 * 2x + 2x * 2x)) = 128 (108 + 24x + (18x + 4x2)) = 128 Combine like terms: 24x + 18x = 42x (108 + 42x + 4x2) = 128 Solving 108 + 42x + 4x2 = 128 Solving for variable 'x'. Reorder the terms: 108 + -128 + 42x + 4x2 = 128 + -128 Combine like terms: 108 + -128 = -20 -20 + 42x + 4x2 = 128 + -128 Combine like terms: 128 + -128 = 0 -20 + 42x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-10 + 21x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-10 + 21x + 2x2)' equal to zero and attempt to solve: Simplifying -10 + 21x + 2x2 = 0 Solving -10 + 21x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5 + 10.5x + x2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 10.5x + 5 + x2 = 0 + 5 Reorder the terms: -5 + 5 + 10.5x + x2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 10.5x + x2 = 0 + 5 10.5x + x2 = 0 + 5 Combine like terms: 0 + 5 = 5 10.5x + x2 = 5 The x term is 10.5x. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5x + 27.5625 + x2 = 5 + 27.5625 Reorder the terms: 27.5625 + 10.5x + x2 = 5 + 27.5625 Combine like terms: 5 + 27.5625 = 32.5625 27.5625 + 10.5x + x2 = 32.5625 Factor a perfect square on the left side: (x + 5.25)(x + 5.25) = 32.5625 Calculate the square root of the right side: 5.706356105 Break this problem into two subproblems by setting (x + 5.25) equal to 5.706356105 and -5.706356105.Subproblem 1
x + 5.25 = 5.706356105 Simplifying x + 5.25 = 5.706356105 Reorder the terms: 5.25 + x = 5.706356105 Solving 5.25 + x = 5.706356105 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = 5.706356105 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = 5.706356105 + -5.25 x = 5.706356105 + -5.25 Combine like terms: 5.706356105 + -5.25 = 0.456356105 x = 0.456356105 Simplifying x = 0.456356105Subproblem 2
x + 5.25 = -5.706356105 Simplifying x + 5.25 = -5.706356105 Reorder the terms: 5.25 + x = -5.706356105 Solving 5.25 + x = -5.706356105 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = -5.706356105 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = -5.706356105 + -5.25 x = -5.706356105 + -5.25 Combine like terms: -5.706356105 + -5.25 = -10.956356105 x = -10.956356105 Simplifying x = -10.956356105Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.456356105, -10.956356105}Solution
x = {0.456356105, -10.956356105}
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