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Simplifying (x + 4)(x + 4) + (4x + 1)(4x + 1) = 49 Reorder the terms: (4 + x)(x + 4) + (4x + 1)(4x + 1) = 49 Reorder the terms: (4 + x)(4 + x) + (4x + 1)(4x + 1) = 49 Multiply (4 + x) * (4 + x) (4(4 + x) + x(4 + x)) + (4x + 1)(4x + 1) = 49 ((4 * 4 + x * 4) + x(4 + x)) + (4x + 1)(4x + 1) = 49 ((16 + 4x) + x(4 + x)) + (4x + 1)(4x + 1) = 49 (16 + 4x + (4 * x + x * x)) + (4x + 1)(4x + 1) = 49 (16 + 4x + (4x + x2)) + (4x + 1)(4x + 1) = 49 Combine like terms: 4x + 4x = 8x (16 + 8x + x2) + (4x + 1)(4x + 1) = 49 Reorder the terms: 16 + 8x + x2 + (1 + 4x)(4x + 1) = 49 Reorder the terms: 16 + 8x + x2 + (1 + 4x)(1 + 4x) = 49 Multiply (1 + 4x) * (1 + 4x) 16 + 8x + x2 + (1(1 + 4x) + 4x * (1 + 4x)) = 49 16 + 8x + x2 + ((1 * 1 + 4x * 1) + 4x * (1 + 4x)) = 49 16 + 8x + x2 + ((1 + 4x) + 4x * (1 + 4x)) = 49 16 + 8x + x2 + (1 + 4x + (1 * 4x + 4x * 4x)) = 49 16 + 8x + x2 + (1 + 4x + (4x + 16x2)) = 49 Combine like terms: 4x + 4x = 8x 16 + 8x + x2 + (1 + 8x + 16x2) = 49 Reorder the terms: 16 + 1 + 8x + 8x + x2 + 16x2 = 49 Combine like terms: 16 + 1 = 17 17 + 8x + 8x + x2 + 16x2 = 49 Combine like terms: 8x + 8x = 16x 17 + 16x + x2 + 16x2 = 49 Combine like terms: x2 + 16x2 = 17x2 17 + 16x + 17x2 = 49 Solving 17 + 16x + 17x2 = 49 Solving for variable 'x'. Reorder the terms: 17 + -49 + 16x + 17x2 = 49 + -49 Combine like terms: 17 + -49 = -32 -32 + 16x + 17x2 = 49 + -49 Combine like terms: 49 + -49 = 0 -32 + 16x + 17x2 = 0 Begin completing the square. Divide all terms by 17 the coefficient of the squared term: Divide each side by '17'. -1.882352941 + 0.9411764706x + x2 = 0 Move the constant term to the right: Add '1.882352941' to each side of the equation. -1.882352941 + 0.9411764706x + 1.882352941 + x2 = 0 + 1.882352941 Reorder the terms: -1.882352941 + 1.882352941 + 0.9411764706x + x2 = 0 + 1.882352941 Combine like terms: -1.882352941 + 1.882352941 = 0.000000000 0.000000000 + 0.9411764706x + x2 = 0 + 1.882352941 0.9411764706x + x2 = 0 + 1.882352941 Combine like terms: 0 + 1.882352941 = 1.882352941 0.9411764706x + x2 = 1.882352941 The x term is 0.9411764706x. Take half its coefficient (0.4705882353). Square it (0.2214532872) and add it to both sides. Add '0.2214532872' to each side of the equation. 0.9411764706x + 0.2214532872 + x2 = 1.882352941 + 0.2214532872 Reorder the terms: 0.2214532872 + 0.9411764706x + x2 = 1.882352941 + 0.2214532872 Combine like terms: 1.882352941 + 0.2214532872 = 2.1038062282 0.2214532872 + 0.9411764706x + x2 = 2.1038062282 Factor a perfect square on the left side: (x + 0.4705882353)(x + 0.4705882353) = 2.1038062282 Calculate the square root of the right side: 1.450450354 Break this problem into two subproblems by setting (x + 0.4705882353) equal to 1.450450354 and -1.450450354.Subproblem 1
x + 0.4705882353 = 1.450450354 Simplifying x + 0.4705882353 = 1.450450354 Reorder the terms: 0.4705882353 + x = 1.450450354 Solving 0.4705882353 + x = 1.450450354 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4705882353' to each side of the equation. 0.4705882353 + -0.4705882353 + x = 1.450450354 + -0.4705882353 Combine like terms: 0.4705882353 + -0.4705882353 = 0.0000000000 0.0000000000 + x = 1.450450354 + -0.4705882353 x = 1.450450354 + -0.4705882353 Combine like terms: 1.450450354 + -0.4705882353 = 0.9798621187 x = 0.9798621187 Simplifying x = 0.9798621187Subproblem 2
x + 0.4705882353 = -1.450450354 Simplifying x + 0.4705882353 = -1.450450354 Reorder the terms: 0.4705882353 + x = -1.450450354 Solving 0.4705882353 + x = -1.450450354 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4705882353' to each side of the equation. 0.4705882353 + -0.4705882353 + x = -1.450450354 + -0.4705882353 Combine like terms: 0.4705882353 + -0.4705882353 = 0.0000000000 0.0000000000 + x = -1.450450354 + -0.4705882353 x = -1.450450354 + -0.4705882353 Combine like terms: -1.450450354 + -0.4705882353 = -1.9210385893 x = -1.9210385893 Simplifying x = -1.9210385893Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9798621187, -1.9210385893}
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