4x^2=x^2-14x-5

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Solution for 4x^2=x^2-14x-5 equation:


Simplifying
4x2 = x2 + -14x + -5

Reorder the terms:
4x2 = -5 + -14x + x2

Solving
4x2 = -5 + -14x + x2

Solving for variable 'x'.

Reorder the terms:
5 + 14x + 4x2 + -1x2 = -5 + -14x + x2 + 5 + 14x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
5 + 14x + 3x2 = -5 + -14x + x2 + 5 + 14x + -1x2

Reorder the terms:
5 + 14x + 3x2 = -5 + 5 + -14x + 14x + x2 + -1x2

Combine like terms: -5 + 5 = 0
5 + 14x + 3x2 = 0 + -14x + 14x + x2 + -1x2
5 + 14x + 3x2 = -14x + 14x + x2 + -1x2

Combine like terms: -14x + 14x = 0
5 + 14x + 3x2 = 0 + x2 + -1x2
5 + 14x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
5 + 14x + 3x2 = 0

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

Divide each side by '3'.
1.666666667 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 4.666666667x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 4.666666667x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 4.666666667x + x2 = 0 + -1.666666667
4.666666667x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
4.666666667x + x2 = -1.666666667

The x term is 4.666666667x.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667x + 5.444444448 + x2 = -1.666666667 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = -1.666666667 + 5.444444448

Combine like terms: -1.666666667 + 5.444444448 = 3.777777781
5.444444448 + 4.666666667x + x2 = 3.777777781

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 3.777777781

Calculate the square root of the right side: 1.943650632

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 1.943650632 and -1.943650632.

Subproblem 1

x + 2.333333334 = 1.943650632 Simplifying x + 2.333333334 = 1.943650632 Reorder the terms: 2.333333334 + x = 1.943650632 Solving 2.333333334 + x = 1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = 1.943650632 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = 1.943650632 + -2.333333334 x = 1.943650632 + -2.333333334 Combine like terms: 1.943650632 + -2.333333334 = -0.389682702 x = -0.389682702 Simplifying x = -0.389682702

Subproblem 2

x + 2.333333334 = -1.943650632 Simplifying x + 2.333333334 = -1.943650632 Reorder the terms: 2.333333334 + x = -1.943650632 Solving 2.333333334 + x = -1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = -1.943650632 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = -1.943650632 + -2.333333334 x = -1.943650632 + -2.333333334 Combine like terms: -1.943650632 + -2.333333334 = -4.276983966 x = -4.276983966 Simplifying x = -4.276983966

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.389682702, -4.276983966}

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