(15x+42)10x-7=5x+3

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Solution for (15x+42)10x-7=5x+3 equation: 


Simplifying
(15x + 42) * 10x + -7 = 5x + 3

Reorder the terms:
(42 + 15x) * 10x + -7 = 5x + 3

Reorder the terms for easier multiplication:
10x(42 + 15x) + -7 = 5x + 3
(42 * 10x + 15x * 10x) + -7 = 5x + 3
(420x + 150x2) + -7 = 5x + 3

Reorder the terms:
-7 + 420x + 150x2 = 5x + 3

Reorder the terms:
-7 + 420x + 150x2 = 3 + 5x

Solving
-7 + 420x + 150x2 = 3 + 5x

Solving for variable 'x'.

Reorder the terms:
-7 + -3 + 420x + -5x + 150x2 = 3 + 5x + -3 + -5x

Combine like terms: -7 + -3 = -10
-10 + 420x + -5x + 150x2 = 3 + 5x + -3 + -5x

Combine like terms: 420x + -5x = 415x
-10 + 415x + 150x2 = 3 + 5x + -3 + -5x

Reorder the terms:
-10 + 415x + 150x2 = 3 + -3 + 5x + -5x

Combine like terms: 3 + -3 = 0
-10 + 415x + 150x2 = 0 + 5x + -5x
-10 + 415x + 150x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-10 + 415x + 150x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-2 + 83x + 30x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-2 + 83x + 30x2)' equal to zero and attempt to solve:

Simplifying
-2 + 83x + 30x2 = 0

Solving
-2 + 83x + 30x2 = 0

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

Divide each side by '30'.
-0.06666666667 + 2.766666667x + x2 = 0.0

Move the constant term to the right:

Add '0.06666666667' to each side of the equation.
-0.06666666667 + 2.766666667x + 0.06666666667 + x2 = 0.0 + 0.06666666667

Reorder the terms:
-0.06666666667 + 0.06666666667 + 2.766666667x + x2 = 0.0 + 0.06666666667

Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000
0.00000000000 + 2.766666667x + x2 = 0.0 + 0.06666666667
2.766666667x + x2 = 0.0 + 0.06666666667

Combine like terms: 0.0 + 0.06666666667 = 0.06666666667
2.766666667x + x2 = 0.06666666667

The x term is 2.766666667x.  Take half its coefficient (1.383333334).
Square it (1.913611113) and add it to both sides.

Add '1.913611113' to each side of the equation.
2.766666667x + 1.913611113 + x2 = 0.06666666667 + 1.913611113

Reorder the terms:
1.913611113 + 2.766666667x + x2 = 0.06666666667 + 1.913611113

Combine like terms: 0.06666666667 + 1.913611113 = 1.98027777967
1.913611113 + 2.766666667x + x2 = 1.98027777967

Factor a perfect square on the left side:
(x + 1.383333334)(x + 1.383333334) = 1.98027777967

Calculate the square root of the right side: 1.407223429

Break this problem into two subproblems by setting 
(x + 1.383333334) equal to 1.407223429 and -1.407223429.


Subproblem 1
x + 1.383333334 = 1.407223429

Simplifying
x + 1.383333334 = 1.407223429

Reorder the terms:
1.383333334 + x = 1.407223429

Solving
1.383333334 + x = 1.407223429

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.383333334' to each side of the equation.
1.383333334 + -1.383333334 + x = 1.407223429 + -1.383333334

Combine like terms: 1.383333334 + -1.383333334 = 0.000000000
0.000000000 + x = 1.407223429 + -1.383333334
x = 1.407223429 + -1.383333334

Combine like terms: 1.407223429 + -1.383333334 = 0.023890095
x = 0.023890095

Simplifying
x = 0.023890095


Subproblem 2
x + 1.383333334 = -1.407223429

Simplifying
x + 1.383333334 = -1.407223429

Reorder the terms:
1.383333334 + x = -1.407223429

Solving
1.383333334 + x = -1.407223429

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.383333334' to each side of the equation.
1.383333334 + -1.383333334 + x = -1.407223429 + -1.383333334

Combine like terms: 1.383333334 + -1.383333334 = 0.000000000
0.000000000 + x = -1.407223429 + -1.383333334
x = -1.407223429 + -1.383333334

Combine like terms: -1.407223429 + -1.383333334 = -2.790556763
x = -2.790556763

Simplifying
x = -2.790556763


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.023890095, -2.790556763}
Solution
x = {0.023890095, -2.790556763}

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