4x+7=14x^2+6x-4

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Solution for 4x+7=14x^2+6x-4 equation:


Simplifying
4x + 7 = 14x2 + 6x + -4

Reorder the terms:
7 + 4x = 14x2 + 6x + -4

Reorder the terms:
7 + 4x = -4 + 6x + 14x2

Solving
7 + 4x = -4 + 6x + 14x2

Solving for variable 'x'.

Reorder the terms:
7 + 4 + 4x + -6x + -14x2 = -4 + 6x + 14x2 + 4 + -6x + -14x2

Combine like terms: 7 + 4 = 11
11 + 4x + -6x + -14x2 = -4 + 6x + 14x2 + 4 + -6x + -14x2

Combine like terms: 4x + -6x = -2x
11 + -2x + -14x2 = -4 + 6x + 14x2 + 4 + -6x + -14x2

Reorder the terms:
11 + -2x + -14x2 = -4 + 4 + 6x + -6x + 14x2 + -14x2

Combine like terms: -4 + 4 = 0
11 + -2x + -14x2 = 0 + 6x + -6x + 14x2 + -14x2
11 + -2x + -14x2 = 6x + -6x + 14x2 + -14x2

Combine like terms: 6x + -6x = 0
11 + -2x + -14x2 = 0 + 14x2 + -14x2
11 + -2x + -14x2 = 14x2 + -14x2

Combine like terms: 14x2 + -14x2 = 0
11 + -2x + -14x2 = 0

Begin completing the square.  Divide all terms by
-14 the coefficient of the squared term: 

Divide each side by '-14'.
-0.7857142857 + 0.1428571429x + x2 = 0

Move the constant term to the right:

Add '0.7857142857' to each side of the equation.
-0.7857142857 + 0.1428571429x + 0.7857142857 + x2 = 0 + 0.7857142857

Reorder the terms:
-0.7857142857 + 0.7857142857 + 0.1428571429x + x2 = 0 + 0.7857142857

Combine like terms: -0.7857142857 + 0.7857142857 = 0.0000000000
0.0000000000 + 0.1428571429x + x2 = 0 + 0.7857142857
0.1428571429x + x2 = 0 + 0.7857142857

Combine like terms: 0 + 0.7857142857 = 0.7857142857
0.1428571429x + x2 = 0.7857142857

The x term is 0.1428571429x.  Take half its coefficient (0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
0.1428571429x + 0.005102040819 + x2 = 0.7857142857 + 0.005102040819

Reorder the terms:
0.005102040819 + 0.1428571429x + x2 = 0.7857142857 + 0.005102040819

Combine like terms: 0.7857142857 + 0.005102040819 = 0.790816326519
0.005102040819 + 0.1428571429x + x2 = 0.790816326519

Factor a perfect square on the left side:
(x + 0.07142857145)(x + 0.07142857145) = 0.790816326519

Calculate the square root of the right side: 0.889278543

Break this problem into two subproblems by setting 
(x + 0.07142857145) equal to 0.889278543 and -0.889278543.

Subproblem 1

x + 0.07142857145 = 0.889278543 Simplifying x + 0.07142857145 = 0.889278543 Reorder the terms: 0.07142857145 + x = 0.889278543 Solving 0.07142857145 + x = 0.889278543 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + x = 0.889278543 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + x = 0.889278543 + -0.07142857145 x = 0.889278543 + -0.07142857145 Combine like terms: 0.889278543 + -0.07142857145 = 0.81784997155 x = 0.81784997155 Simplifying x = 0.81784997155

Subproblem 2

x + 0.07142857145 = -0.889278543 Simplifying x + 0.07142857145 = -0.889278543 Reorder the terms: 0.07142857145 + x = -0.889278543 Solving 0.07142857145 + x = -0.889278543 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + x = -0.889278543 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + x = -0.889278543 + -0.07142857145 x = -0.889278543 + -0.07142857145 Combine like terms: -0.889278543 + -0.07142857145 = -0.96070711445 x = -0.96070711445 Simplifying x = -0.96070711445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.81784997155, -0.96070711445}

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