(x+4)(x+4)+(4x+1)(4x+1)=49

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Solution for (x+4)(x+4)+(4x+1)(4x+1)=49 equation:


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 = 0.9798621187

Subproblem 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 = -1.9210385893

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.9798621187, -1.9210385893}

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