6x^2-7x-222=0

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Solution for 6x^2-7x-222=0 equation:


Simplifying
6x2 + -7x + -222 = 0

Reorder the terms:
-222 + -7x + 6x2 = 0

Solving
-222 + -7x + 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'.
-37 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '37' to each side of the equation.
-37 + -1.166666667x + 37 + x2 = 0 + 37

Reorder the terms:
-37 + 37 + -1.166666667x + x2 = 0 + 37

Combine like terms: -37 + 37 = 0
0 + -1.166666667x + x2 = 0 + 37
-1.166666667x + x2 = 0 + 37

Combine like terms: 0 + 37 = 37
-1.166666667x + x2 = 37

The x term is -1.166666667x.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667x + 0.3402777780 + x2 = 37 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 37 + 0.3402777780

Combine like terms: 37 + 0.3402777780 = 37.340277778
0.3402777780 + -1.166666667x + x2 = 37.340277778

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 37.340277778

Calculate the square root of the right side: 6.110669176

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 6.110669176 and -6.110669176.

Subproblem 1

x + -0.5833333335 = 6.110669176 Simplifying x + -0.5833333335 = 6.110669176 Reorder the terms: -0.5833333335 + x = 6.110669176 Solving -0.5833333335 + x = 6.110669176 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = 6.110669176 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 6.110669176 + 0.5833333335 x = 6.110669176 + 0.5833333335 Combine like terms: 6.110669176 + 0.5833333335 = 6.6940025095 x = 6.6940025095 Simplifying x = 6.6940025095

Subproblem 2

x + -0.5833333335 = -6.110669176 Simplifying x + -0.5833333335 = -6.110669176 Reorder the terms: -0.5833333335 + x = -6.110669176 Solving -0.5833333335 + x = -6.110669176 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = -6.110669176 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -6.110669176 + 0.5833333335 x = -6.110669176 + 0.5833333335 Combine like terms: -6.110669176 + 0.5833333335 = -5.5273358425 x = -5.5273358425 Simplifying x = -5.5273358425

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.6940025095, -5.5273358425}

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