7x(x+3)-5x=7

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Solution for 7x(x+3)-5x=7 equation:


Simplifying
7x(x + 3) + -5x = 7

Reorder the terms:
7x(3 + x) + -5x = 7
(3 * 7x + x * 7x) + -5x = 7
(21x + 7x2) + -5x = 7

Reorder the terms:
21x + -5x + 7x2 = 7

Combine like terms: 21x + -5x = 16x
16x + 7x2 = 7

Solving
16x + 7x2 = 7

Solving for variable 'x'.

Reorder the terms:
-7 + 16x + 7x2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 16x + 7x2 = 0

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

Divide each side by '7'.
-1 + 2.285714286x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2.285714286x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2.285714286x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2.285714286x + x2 = 0 + 1
2.285714286x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
2.285714286x + x2 = 1

The x term is 2.285714286x.  Take half its coefficient (1.142857143).
Square it (1.306122449) and add it to both sides.

Add '1.306122449' to each side of the equation.
2.285714286x + 1.306122449 + x2 = 1 + 1.306122449

Reorder the terms:
1.306122449 + 2.285714286x + x2 = 1 + 1.306122449

Combine like terms: 1 + 1.306122449 = 2.306122449
1.306122449 + 2.285714286x + x2 = 2.306122449

Factor a perfect square on the left side:
(x + 1.142857143)(x + 1.142857143) = 2.306122449

Calculate the square root of the right side: 1.518592259

Break this problem into two subproblems by setting 
(x + 1.142857143) equal to 1.518592259 and -1.518592259.

Subproblem 1

x + 1.142857143 = 1.518592259 Simplifying x + 1.142857143 = 1.518592259 Reorder the terms: 1.142857143 + x = 1.518592259 Solving 1.142857143 + x = 1.518592259 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + x = 1.518592259 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + x = 1.518592259 + -1.142857143 x = 1.518592259 + -1.142857143 Combine like terms: 1.518592259 + -1.142857143 = 0.375735116 x = 0.375735116 Simplifying x = 0.375735116

Subproblem 2

x + 1.142857143 = -1.518592259 Simplifying x + 1.142857143 = -1.518592259 Reorder the terms: 1.142857143 + x = -1.518592259 Solving 1.142857143 + x = -1.518592259 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + x = -1.518592259 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + x = -1.518592259 + -1.142857143 x = -1.518592259 + -1.142857143 Combine like terms: -1.518592259 + -1.142857143 = -2.661449402 x = -2.661449402 Simplifying x = -2.661449402

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.375735116, -2.661449402}

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