7x+5=(2x+3)(4x+1)

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Solution for 7x+5=(2x+3)(4x+1) equation:


Simplifying
7x + 5 = (2x + 3)(4x + 1)

Reorder the terms:
5 + 7x = (2x + 3)(4x + 1)

Reorder the terms:
5 + 7x = (3 + 2x)(4x + 1)

Reorder the terms:
5 + 7x = (3 + 2x)(1 + 4x)

Multiply (3 + 2x) * (1 + 4x)
5 + 7x = (3(1 + 4x) + 2x * (1 + 4x))
5 + 7x = ((1 * 3 + 4x * 3) + 2x * (1 + 4x))
5 + 7x = ((3 + 12x) + 2x * (1 + 4x))
5 + 7x = (3 + 12x + (1 * 2x + 4x * 2x))
5 + 7x = (3 + 12x + (2x + 8x2))

Combine like terms: 12x + 2x = 14x
5 + 7x = (3 + 14x + 8x2)

Solving
5 + 7x = 3 + 14x + 8x2

Solving for variable 'x'.

Reorder the terms:
5 + -3 + 7x + -14x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2

Combine like terms: 5 + -3 = 2
2 + 7x + -14x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2

Combine like terms: 7x + -14x = -7x
2 + -7x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2

Reorder the terms:
2 + -7x + -8x2 = 3 + -3 + 14x + -14x + 8x2 + -8x2

Combine like terms: 3 + -3 = 0
2 + -7x + -8x2 = 0 + 14x + -14x + 8x2 + -8x2
2 + -7x + -8x2 = 14x + -14x + 8x2 + -8x2

Combine like terms: 14x + -14x = 0
2 + -7x + -8x2 = 0 + 8x2 + -8x2
2 + -7x + -8x2 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
2 + -7x + -8x2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-0.25 + 0.875x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.875x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.875x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.875x + x2 = 0 + 0.25
0.875x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.875x + x2 = 0.25

The x term is 0.875x.  Take half its coefficient (0.4375).
Square it (0.19140625) and add it to both sides.

Add '0.19140625' to each side of the equation.
0.875x + 0.19140625 + x2 = 0.25 + 0.19140625

Reorder the terms:
0.19140625 + 0.875x + x2 = 0.25 + 0.19140625

Combine like terms: 0.25 + 0.19140625 = 0.44140625
0.19140625 + 0.875x + x2 = 0.44140625

Factor a perfect square on the left side:
(x + 0.4375)(x + 0.4375) = 0.44140625

Calculate the square root of the right side: 0.664384113

Break this problem into two subproblems by setting 
(x + 0.4375) equal to 0.664384113 and -0.664384113.

Subproblem 1

x + 0.4375 = 0.664384113 Simplifying x + 0.4375 = 0.664384113 Reorder the terms: 0.4375 + x = 0.664384113 Solving 0.4375 + x = 0.664384113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = 0.664384113 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = 0.664384113 + -0.4375 x = 0.664384113 + -0.4375 Combine like terms: 0.664384113 + -0.4375 = 0.226884113 x = 0.226884113 Simplifying x = 0.226884113

Subproblem 2

x + 0.4375 = -0.664384113 Simplifying x + 0.4375 = -0.664384113 Reorder the terms: 0.4375 + x = -0.664384113 Solving 0.4375 + x = -0.664384113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = -0.664384113 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = -0.664384113 + -0.4375 x = -0.664384113 + -0.4375 Combine like terms: -0.664384113 + -0.4375 = -1.101884113 x = -1.101884113 Simplifying x = -1.101884113

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.226884113, -1.101884113}

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