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Simplifying (14 + 2x)(30 + 2x) + -420 + 4x = 960 Multiply (14 + 2x) * (30 + 2x) (14(30 + 2x) + 2x * (30 + 2x)) + -420 + 4x = 960 ((30 * 14 + 2x * 14) + 2x * (30 + 2x)) + -420 + 4x = 960 ((420 + 28x) + 2x * (30 + 2x)) + -420 + 4x = 960 (420 + 28x + (30 * 2x + 2x * 2x)) + -420 + 4x = 960 (420 + 28x + (60x + 4x2)) + -420 + 4x = 960 Combine like terms: 28x + 60x = 88x (420 + 88x + 4x2) + -420 + 4x = 960 Reorder the terms: 420 + -420 + 88x + 4x + 4x2 = 960 Combine like terms: 420 + -420 = 0 0 + 88x + 4x + 4x2 = 960 88x + 4x + 4x2 = 960 Combine like terms: 88x + 4x = 92x 92x + 4x2 = 960 Solving 92x + 4x2 = 960 Solving for variable 'x'. Reorder the terms: -960 + 92x + 4x2 = 960 + -960 Combine like terms: 960 + -960 = 0 -960 + 92x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-240 + 23x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-240 + 23x + x2)' equal to zero and attempt to solve: Simplifying -240 + 23x + x2 = 0 Solving -240 + 23x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '240' to each side of the equation. -240 + 23x + 240 + x2 = 0 + 240 Reorder the terms: -240 + 240 + 23x + x2 = 0 + 240 Combine like terms: -240 + 240 = 0 0 + 23x + x2 = 0 + 240 23x + x2 = 0 + 240 Combine like terms: 0 + 240 = 240 23x + x2 = 240 The x term is 23x. Take half its coefficient (11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. 23x + 132.25 + x2 = 240 + 132.25 Reorder the terms: 132.25 + 23x + x2 = 240 + 132.25 Combine like terms: 240 + 132.25 = 372.25 132.25 + 23x + x2 = 372.25 Factor a perfect square on the left side: (x + 11.5)(x + 11.5) = 372.25 Calculate the square root of the right side: 19.293781382 Break this problem into two subproblems by setting (x + 11.5) equal to 19.293781382 and -19.293781382.Subproblem 1
x + 11.5 = 19.293781382 Simplifying x + 11.5 = 19.293781382 Reorder the terms: 11.5 + x = 19.293781382 Solving 11.5 + x = 19.293781382 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = 19.293781382 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = 19.293781382 + -11.5 x = 19.293781382 + -11.5 Combine like terms: 19.293781382 + -11.5 = 7.793781382 x = 7.793781382 Simplifying x = 7.793781382Subproblem 2
x + 11.5 = -19.293781382 Simplifying x + 11.5 = -19.293781382 Reorder the terms: 11.5 + x = -19.293781382 Solving 11.5 + x = -19.293781382 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + x = -19.293781382 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + x = -19.293781382 + -11.5 x = -19.293781382 + -11.5 Combine like terms: -19.293781382 + -11.5 = -30.793781382 x = -30.793781382 Simplifying x = -30.793781382Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.793781382, -30.793781382}Solution
x = {7.793781382, -30.793781382}
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