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Simplifying 2(x + 1) * 2(x + 1) + 2(x + 1) + 2(x + 1) = 5(x + 10) Reorder the terms: 2(1 + x) * 2(x + 1) + 2(x + 1) + 2(x + 1) = 5(x + 10) Reorder the terms: 2(1 + x) * 2(1 + x) + 2(x + 1) + 2(x + 1) = 5(x + 10) Reorder the terms for easier multiplication: 2 * 2(1 + x)(1 + x) + 2(x + 1) + 2(x + 1) = 5(x + 10) Multiply 2 * 2 4(1 + x)(1 + x) + 2(x + 1) + 2(x + 1) = 5(x + 10) Multiply (1 + x) * (1 + x) 4(1(1 + x) + x(1 + x)) + 2(x + 1) + 2(x + 1) = 5(x + 10) 4((1 * 1 + x * 1) + x(1 + x)) + 2(x + 1) + 2(x + 1) = 5(x + 10) 4((1 + 1x) + x(1 + x)) + 2(x + 1) + 2(x + 1) = 5(x + 10) 4(1 + 1x + (1 * x + x * x)) + 2(x + 1) + 2(x + 1) = 5(x + 10) 4(1 + 1x + (1x + x2)) + 2(x + 1) + 2(x + 1) = 5(x + 10) Combine like terms: 1x + 1x = 2x 4(1 + 2x + x2) + 2(x + 1) + 2(x + 1) = 5(x + 10) (1 * 4 + 2x * 4 + x2 * 4) + 2(x + 1) + 2(x + 1) = 5(x + 10) (4 + 8x + 4x2) + 2(x + 1) + 2(x + 1) = 5(x + 10) Reorder the terms: 4 + 8x + 4x2 + 2(1 + x) + 2(x + 1) = 5(x + 10) 4 + 8x + 4x2 + (1 * 2 + x * 2) + 2(x + 1) = 5(x + 10) 4 + 8x + 4x2 + (2 + 2x) + 2(x + 1) = 5(x + 10) Reorder the terms: 4 + 8x + 4x2 + 2 + 2x + 2(1 + x) = 5(x + 10) 4 + 8x + 4x2 + 2 + 2x + (1 * 2 + x * 2) = 5(x + 10) 4 + 8x + 4x2 + 2 + 2x + (2 + 2x) = 5(x + 10) Reorder the terms: 4 + 2 + 2 + 8x + 2x + 2x + 4x2 = 5(x + 10) Combine like terms: 4 + 2 = 6 6 + 2 + 8x + 2x + 2x + 4x2 = 5(x + 10) Combine like terms: 6 + 2 = 8 8 + 8x + 2x + 2x + 4x2 = 5(x + 10) Combine like terms: 8x + 2x = 10x 8 + 10x + 2x + 4x2 = 5(x + 10) Combine like terms: 10x + 2x = 12x 8 + 12x + 4x2 = 5(x + 10) Reorder the terms: 8 + 12x + 4x2 = 5(10 + x) 8 + 12x + 4x2 = (10 * 5 + x * 5) 8 + 12x + 4x2 = (50 + 5x) Solving 8 + 12x + 4x2 = 50 + 5x Solving for variable 'x'. Reorder the terms: 8 + -50 + 12x + -5x + 4x2 = 50 + 5x + -50 + -5x Combine like terms: 8 + -50 = -42 -42 + 12x + -5x + 4x2 = 50 + 5x + -50 + -5x Combine like terms: 12x + -5x = 7x -42 + 7x + 4x2 = 50 + 5x + -50 + -5x Reorder the terms: -42 + 7x + 4x2 = 50 + -50 + 5x + -5x Combine like terms: 50 + -50 = 0 -42 + 7x + 4x2 = 0 + 5x + -5x -42 + 7x + 4x2 = 5x + -5x Combine like terms: 5x + -5x = 0 -42 + 7x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -10.5 + 1.75x + x2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + 1.75x + 10.5 + x2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + 1.75x + x2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + 1.75x + x2 = 0 + 10.5 1.75x + x2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 1.75x + x2 = 10.5 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = 10.5 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = 10.5 + 0.765625 Combine like terms: 10.5 + 0.765625 = 11.265625 0.765625 + 1.75x + x2 = 11.265625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 11.265625 Calculate the square root of the right side: 3.356430396 Break this problem into two subproblems by setting (x + 0.875) equal to 3.356430396 and -3.356430396.Subproblem 1
x + 0.875 = 3.356430396 Simplifying x + 0.875 = 3.356430396 Reorder the terms: 0.875 + x = 3.356430396 Solving 0.875 + x = 3.356430396 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 3.356430396 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 3.356430396 + -0.875 x = 3.356430396 + -0.875 Combine like terms: 3.356430396 + -0.875 = 2.481430396 x = 2.481430396 Simplifying x = 2.481430396Subproblem 2
x + 0.875 = -3.356430396 Simplifying x + 0.875 = -3.356430396 Reorder the terms: 0.875 + x = -3.356430396 Solving 0.875 + x = -3.356430396 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -3.356430396 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -3.356430396 + -0.875 x = -3.356430396 + -0.875 Combine like terms: -3.356430396 + -0.875 = -4.231430396 x = -4.231430396 Simplifying x = -4.231430396Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.481430396, -4.231430396}
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