6x^2+1x-23=0

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Solution for 6x^2+1x-23=0 equation:


Simplifying
6x2 + 1x + -23 = 0

Reorder the terms:
-23 + 1x + 6x2 = 0

Solving
-23 + 1x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.833333333 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + 0.1666666667x + 3.833333333 + x2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + 0.1666666667x + x2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + 0.1666666667x + x2 = 0 + 3.833333333
0.1666666667x + x2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
0.1666666667x + x2 = 3.833333333

The x term is 0.1666666667x.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667x + 0.006944444447 + x2 = 3.833333333 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667x + x2 = 3.833333333 + 0.006944444447

Combine like terms: 3.833333333 + 0.006944444447 = 3.840277777447
0.006944444447 + 0.1666666667x + x2 = 3.840277777447

Factor a perfect square on the left side:
(x + 0.08333333335)(x + 0.08333333335) = 3.840277777447

Calculate the square root of the right side: 1.959662669

Break this problem into two subproblems by setting 
(x + 0.08333333335) equal to 1.959662669 and -1.959662669.

Subproblem 1

x + 0.08333333335 = 1.959662669 Simplifying x + 0.08333333335 = 1.959662669 Reorder the terms: 0.08333333335 + x = 1.959662669 Solving 0.08333333335 + x = 1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 1.959662669 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 1.959662669 + -0.08333333335 x = 1.959662669 + -0.08333333335 Combine like terms: 1.959662669 + -0.08333333335 = 1.87632933565 x = 1.87632933565 Simplifying x = 1.87632933565

Subproblem 2

x + 0.08333333335 = -1.959662669 Simplifying x + 0.08333333335 = -1.959662669 Reorder the terms: 0.08333333335 + x = -1.959662669 Solving 0.08333333335 + x = -1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -1.959662669 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -1.959662669 + -0.08333333335 x = -1.959662669 + -0.08333333335 Combine like terms: -1.959662669 + -0.08333333335 = -2.04299600235 x = -2.04299600235 Simplifying x = -2.04299600235

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.87632933565, -2.04299600235}

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