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Simplifying (2(x(x + 1))) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 Reorder the terms: (2(x(1 + x))) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 (2((1 * x + x * x))) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 (2((1x + x2))) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 ((1x * 2 + x2 * 2)) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 ((2x + 2x2)) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 (2x + 2x2) + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 Remove parenthesis around (2x + 2x2) 2x + 2x2 + (2(x(x + 5))) + (2((x + 1)(x + 5))) = 172 Reorder the terms: 2x + 2x2 + (2(x(5 + x))) + (2((x + 1)(x + 5))) = 172 2x + 2x2 + (2((5 * x + x * x))) + (2((x + 1)(x + 5))) = 172 2x + 2x2 + (2((5x + x2))) + (2((x + 1)(x + 5))) = 172 2x + 2x2 + ((5x * 2 + x2 * 2)) + (2((x + 1)(x + 5))) = 172 2x + 2x2 + ((10x + 2x2)) + (2((x + 1)(x + 5))) = 172 2x + 2x2 + (10x + 2x2) + (2((x + 1)(x + 5))) = 172 Remove parenthesis around (10x + 2x2) 2x + 2x2 + 10x + 2x2 + (2((x + 1)(x + 5))) = 172 Reorder the terms: 2x + 2x2 + 10x + 2x2 + (2((1 + x)(x + 5))) = 172 Reorder the terms: 2x + 2x2 + 10x + 2x2 + (2((1 + x)(5 + x))) = 172 Multiply (1 + x) * (5 + x) 2x + 2x2 + 10x + 2x2 + (2((1(5 + x) + x(5 + x)))) = 172 2x + 2x2 + 10x + 2x2 + (2(((5 * 1 + x * 1) + x(5 + x)))) = 172 2x + 2x2 + 10x + 2x2 + (2(((5 + 1x) + x(5 + x)))) = 172 2x + 2x2 + 10x + 2x2 + (2((5 + 1x + (5 * x + x * x)))) = 172 2x + 2x2 + 10x + 2x2 + (2((5 + 1x + (5x + x2)))) = 172 Combine like terms: 1x + 5x = 6x 2x + 2x2 + 10x + 2x2 + (2((5 + 6x + x2))) = 172 2x + 2x2 + 10x + 2x2 + ((5 * 2 + 6x * 2 + x2 * 2)) = 172 2x + 2x2 + 10x + 2x2 + ((10 + 12x + 2x2)) = 172 2x + 2x2 + 10x + 2x2 + (10 + 12x + 2x2) = 172 Remove parenthesis around (10 + 12x + 2x2) 2x + 2x2 + 10x + 2x2 + 10 + 12x + 2x2 = 172 Reorder the terms: 10 + 2x + 10x + 12x + 2x2 + 2x2 + 2x2 = 172 Combine like terms: 2x + 10x = 12x 10 + 12x + 12x + 2x2 + 2x2 + 2x2 = 172 Combine like terms: 12x + 12x = 24x 10 + 24x + 2x2 + 2x2 + 2x2 = 172 Combine like terms: 2x2 + 2x2 = 4x2 10 + 24x + 4x2 + 2x2 = 172 Combine like terms: 4x2 + 2x2 = 6x2 10 + 24x + 6x2 = 172 Solving 10 + 24x + 6x2 = 172 Solving for variable 'x'. Reorder the terms: 10 + -172 + 24x + 6x2 = 172 + -172 Combine like terms: 10 + -172 = -162 -162 + 24x + 6x2 = 172 + -172 Combine like terms: 172 + -172 = 0 -162 + 24x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-27 + 4x + x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-27 + 4x + x2)' equal to zero and attempt to solve: Simplifying -27 + 4x + x2 = 0 Solving -27 + 4x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '27' to each side of the equation. -27 + 4x + 27 + x2 = 0 + 27 Reorder the terms: -27 + 27 + 4x + x2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + 4x + x2 = 0 + 27 4x + x2 = 0 + 27 Combine like terms: 0 + 27 = 27 4x + x2 = 27 The x term is 4x. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4x + 4 + x2 = 27 + 4 Reorder the terms: 4 + 4x + x2 = 27 + 4 Combine like terms: 27 + 4 = 31 4 + 4x + x2 = 31 Factor a perfect square on the left side: (x + 2)(x + 2) = 31 Calculate the square root of the right side: 5.567764363 Break this problem into two subproblems by setting (x + 2) equal to 5.567764363 and -5.567764363.Subproblem 1
x + 2 = 5.567764363 Simplifying x + 2 = 5.567764363 Reorder the terms: 2 + x = 5.567764363 Solving 2 + x = 5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 5.567764363 + -2 Combine like terms: 2 + -2 = 0 0 + x = 5.567764363 + -2 x = 5.567764363 + -2 Combine like terms: 5.567764363 + -2 = 3.567764363 x = 3.567764363 Simplifying x = 3.567764363Subproblem 2
x + 2 = -5.567764363 Simplifying x + 2 = -5.567764363 Reorder the terms: 2 + x = -5.567764363 Solving 2 + x = -5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -5.567764363 + -2 Combine like terms: 2 + -2 = 0 0 + x = -5.567764363 + -2 x = -5.567764363 + -2 Combine like terms: -5.567764363 + -2 = -7.567764363 x = -7.567764363 Simplifying x = -7.567764363Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.567764363, -7.567764363}Solution
x = {3.567764363, -7.567764363}
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