6x^2-3x+5+4x-2x^2-6=0

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Solution for 6x^2-3x+5+4x-2x^2-6=0 equation:


Simplifying
6x2 + -3x + 5 + 4x + -2x2 + -6 = 0

Reorder the terms:
5 + -6 + -3x + 4x + 6x2 + -2x2 = 0

Combine like terms: 5 + -6 = -1
-1 + -3x + 4x + 6x2 + -2x2 = 0

Combine like terms: -3x + 4x = 1x
-1 + 1x + 6x2 + -2x2 = 0

Combine like terms: 6x2 + -2x2 = 4x2
-1 + 1x + 4x2 = 0

Solving
-1 + 1x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-0.25 + 0.25x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.25x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.25x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.25x + x2 = 0 + 0.25
0.25x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.25x + x2 = 0.25

The x term is 0.25x.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25x + 0.015625 + x2 = 0.25 + 0.015625

Reorder the terms:
0.015625 + 0.25x + x2 = 0.25 + 0.015625

Combine like terms: 0.25 + 0.015625 = 0.265625
0.015625 + 0.25x + x2 = 0.265625

Factor a perfect square on the left side:
(x + 0.125)(x + 0.125) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(x + 0.125) equal to 0.515388203 and -0.515388203.

Subproblem 1

x + 0.125 = 0.515388203 Simplifying x + 0.125 = 0.515388203 Reorder the terms: 0.125 + x = 0.515388203 Solving 0.125 + x = 0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 0.515388203 + -0.125 x = 0.515388203 + -0.125 Combine like terms: 0.515388203 + -0.125 = 0.390388203 x = 0.390388203 Simplifying x = 0.390388203

Subproblem 2

x + 0.125 = -0.515388203 Simplifying x + 0.125 = -0.515388203 Reorder the terms: 0.125 + x = -0.515388203 Solving 0.125 + x = -0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -0.515388203 + -0.125 x = -0.515388203 + -0.125 Combine like terms: -0.515388203 + -0.125 = -0.640388203 x = -0.640388203 Simplifying x = -0.640388203

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.390388203, -0.640388203}

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