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Simplifying (11 + 2x)(14 + 2x) = 116 Multiply (11 + 2x) * (14 + 2x) (11(14 + 2x) + 2x * (14 + 2x)) = 116 ((14 * 11 + 2x * 11) + 2x * (14 + 2x)) = 116 ((154 + 22x) + 2x * (14 + 2x)) = 116 (154 + 22x + (14 * 2x + 2x * 2x)) = 116 (154 + 22x + (28x + 4x2)) = 116 Combine like terms: 22x + 28x = 50x (154 + 50x + 4x2) = 116 Solving 154 + 50x + 4x2 = 116 Solving for variable 'x'. Reorder the terms: 154 + -116 + 50x + 4x2 = 116 + -116 Combine like terms: 154 + -116 = 38 38 + 50x + 4x2 = 116 + -116 Combine like terms: 116 + -116 = 0 38 + 50x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(19 + 25x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(19 + 25x + 2x2)' equal to zero and attempt to solve: Simplifying 19 + 25x + 2x2 = 0 Solving 19 + 25x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 9.5 + 12.5x + x2 = 0 Move the constant term to the right: Add '-9.5' to each side of the equation. 9.5 + 12.5x + -9.5 + x2 = 0 + -9.5 Reorder the terms: 9.5 + -9.5 + 12.5x + x2 = 0 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + 12.5x + x2 = 0 + -9.5 12.5x + x2 = 0 + -9.5 Combine like terms: 0 + -9.5 = -9.5 12.5x + x2 = -9.5 The x term is 12.5x. Take half its coefficient (6.25). Square it (39.0625) and add it to both sides. Add '39.0625' to each side of the equation. 12.5x + 39.0625 + x2 = -9.5 + 39.0625 Reorder the terms: 39.0625 + 12.5x + x2 = -9.5 + 39.0625 Combine like terms: -9.5 + 39.0625 = 29.5625 39.0625 + 12.5x + x2 = 29.5625 Factor a perfect square on the left side: (x + 6.25)(x + 6.25) = 29.5625 Calculate the square root of the right side: 5.437140793 Break this problem into two subproblems by setting (x + 6.25) equal to 5.437140793 and -5.437140793.Subproblem 1
x + 6.25 = 5.437140793 Simplifying x + 6.25 = 5.437140793 Reorder the terms: 6.25 + x = 5.437140793 Solving 6.25 + x = 5.437140793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.25' to each side of the equation. 6.25 + -6.25 + x = 5.437140793 + -6.25 Combine like terms: 6.25 + -6.25 = 0.00 0.00 + x = 5.437140793 + -6.25 x = 5.437140793 + -6.25 Combine like terms: 5.437140793 + -6.25 = -0.812859207 x = -0.812859207 Simplifying x = -0.812859207Subproblem 2
x + 6.25 = -5.437140793 Simplifying x + 6.25 = -5.437140793 Reorder the terms: 6.25 + x = -5.437140793 Solving 6.25 + x = -5.437140793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.25' to each side of the equation. 6.25 + -6.25 + x = -5.437140793 + -6.25 Combine like terms: 6.25 + -6.25 = 0.00 0.00 + x = -5.437140793 + -6.25 x = -5.437140793 + -6.25 Combine like terms: -5.437140793 + -6.25 = -11.687140793 x = -11.687140793 Simplifying x = -11.687140793Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.812859207, -11.687140793}Solution
x = {-0.812859207, -11.687140793}
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