7x^2-6x-111=0

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Solution for 7x^2-6x-111=0 equation:


Simplifying
7x2 + -6x + -111 = 0

Reorder the terms:
-111 + -6x + 7x2 = 0

Solving
-111 + -6x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
-15.85714286 + -0.8571428571x + x2 = 0

Move the constant term to the right:

Add '15.85714286' to each side of the equation.
-15.85714286 + -0.8571428571x + 15.85714286 + x2 = 0 + 15.85714286

Reorder the terms:
-15.85714286 + 15.85714286 + -0.8571428571x + x2 = 0 + 15.85714286

Combine like terms: -15.85714286 + 15.85714286 = 0.00000000
0.00000000 + -0.8571428571x + x2 = 0 + 15.85714286
-0.8571428571x + x2 = 0 + 15.85714286

Combine like terms: 0 + 15.85714286 = 15.85714286
-0.8571428571x + x2 = 15.85714286

The x term is -0.8571428571x.  Take half its coefficient (-0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
-0.8571428571x + 0.1836734694 + x2 = 15.85714286 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571x + x2 = 15.85714286 + 0.1836734694

Combine like terms: 15.85714286 + 0.1836734694 = 16.0408163294
0.1836734694 + -0.8571428571x + x2 = 16.0408163294

Factor a perfect square on the left side:
(x + -0.4285714286)(x + -0.4285714286) = 16.0408163294

Calculate the square root of the right side: 4.005098791

Break this problem into two subproblems by setting 
(x + -0.4285714286) equal to 4.005098791 and -4.005098791.

Subproblem 1

x + -0.4285714286 = 4.005098791 Simplifying x + -0.4285714286 = 4.005098791 Reorder the terms: -0.4285714286 + x = 4.005098791 Solving -0.4285714286 + x = 4.005098791 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = 4.005098791 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = 4.005098791 + 0.4285714286 x = 4.005098791 + 0.4285714286 Combine like terms: 4.005098791 + 0.4285714286 = 4.4336702196 x = 4.4336702196 Simplifying x = 4.4336702196

Subproblem 2

x + -0.4285714286 = -4.005098791 Simplifying x + -0.4285714286 = -4.005098791 Reorder the terms: -0.4285714286 + x = -4.005098791 Solving -0.4285714286 + x = -4.005098791 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = -4.005098791 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = -4.005098791 + 0.4285714286 x = -4.005098791 + 0.4285714286 Combine like terms: -4.005098791 + 0.4285714286 = -3.5765273624 x = -3.5765273624 Simplifying x = -3.5765273624

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.4336702196, -3.5765273624}

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