(x*7)(x-5)=5x-37

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Solution for (x*7)(x-5)=5x-37 equation:


Simplifying
(x * 7)(x + -5) = 5x + -37

Reorder the terms for easier multiplication:
(7x)(x + -5) = 5x + -37

Remove parenthesis around (7x)
7x(x + -5) = 5x + -37

Reorder the terms:
7x(-5 + x) = 5x + -37
(-5 * 7x + x * 7x) = 5x + -37
(-35x + 7x2) = 5x + -37

Reorder the terms:
-35x + 7x2 = -37 + 5x

Solving
-35x + 7x2 = -37 + 5x

Solving for variable 'x'.

Reorder the terms:
37 + -35x + -5x + 7x2 = -37 + 5x + 37 + -5x

Combine like terms: -35x + -5x = -40x
37 + -40x + 7x2 = -37 + 5x + 37 + -5x

Reorder the terms:
37 + -40x + 7x2 = -37 + 37 + 5x + -5x

Combine like terms: -37 + 37 = 0
37 + -40x + 7x2 = 0 + 5x + -5x
37 + -40x + 7x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
37 + -40x + 7x2 = 0

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

Divide each side by '7'.
5.285714286 + -5.714285714x + x2 = 0

Move the constant term to the right:

Add '-5.285714286' to each side of the equation.
5.285714286 + -5.714285714x + -5.285714286 + x2 = 0 + -5.285714286

Reorder the terms:
5.285714286 + -5.285714286 + -5.714285714x + x2 = 0 + -5.285714286

Combine like terms: 5.285714286 + -5.285714286 = 0.000000000
0.000000000 + -5.714285714x + x2 = 0 + -5.285714286
-5.714285714x + x2 = 0 + -5.285714286

Combine like terms: 0 + -5.285714286 = -5.285714286
-5.714285714x + x2 = -5.285714286

The x term is -5.714285714x.  Take half its coefficient (-2.857142857).
Square it (8.163265305) and add it to both sides.

Add '8.163265305' to each side of the equation.
-5.714285714x + 8.163265305 + x2 = -5.285714286 + 8.163265305

Reorder the terms:
8.163265305 + -5.714285714x + x2 = -5.285714286 + 8.163265305

Combine like terms: -5.285714286 + 8.163265305 = 2.877551019
8.163265305 + -5.714285714x + x2 = 2.877551019

Factor a perfect square on the left side:
(x + -2.857142857)(x + -2.857142857) = 2.877551019

Calculate the square root of the right side: 1.696334583

Break this problem into two subproblems by setting 
(x + -2.857142857) equal to 1.696334583 and -1.696334583.

Subproblem 1

x + -2.857142857 = 1.696334583 Simplifying x + -2.857142857 = 1.696334583 Reorder the terms: -2.857142857 + x = 1.696334583 Solving -2.857142857 + x = 1.696334583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.857142857' to each side of the equation. -2.857142857 + 2.857142857 + x = 1.696334583 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + x = 1.696334583 + 2.857142857 x = 1.696334583 + 2.857142857 Combine like terms: 1.696334583 + 2.857142857 = 4.55347744 x = 4.55347744 Simplifying x = 4.55347744

Subproblem 2

x + -2.857142857 = -1.696334583 Simplifying x + -2.857142857 = -1.696334583 Reorder the terms: -2.857142857 + x = -1.696334583 Solving -2.857142857 + x = -1.696334583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.857142857' to each side of the equation. -2.857142857 + 2.857142857 + x = -1.696334583 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + x = -1.696334583 + 2.857142857 x = -1.696334583 + 2.857142857 Combine like terms: -1.696334583 + 2.857142857 = 1.160808274 x = 1.160808274 Simplifying x = 1.160808274

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.55347744, 1.160808274}

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