6x^2=5x+23

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Solution for 6x^2=5x+23 equation:


Simplifying
6x2 = 5x + 23

Reorder the terms:
6x2 = 23 + 5x

Solving
6x2 = 23 + 5x

Solving for variable 'x'.

Reorder the terms:
-23 + -5x + 6x2 = 23 + 5x + -23 + -5x

Reorder the terms:
-23 + -5x + 6x2 = 23 + -23 + 5x + -5x

Combine like terms: 23 + -23 = 0
-23 + -5x + 6x2 = 0 + 5x + -5x
-23 + -5x + 6x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-23 + -5x + 6x2 = 0

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

Divide each side by '6'.
-3.833333333 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -0.8333333333x + 3.833333333 + x2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -0.8333333333x + x2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -0.8333333333x + x2 = 0 + 3.833333333
-0.8333333333x + x2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-0.8333333333x + x2 = 3.833333333

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 3.833333333 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 3.833333333 + 0.1736111111

Combine like terms: 3.833333333 + 0.1736111111 = 4.0069444441
0.1736111111 + -0.8333333333x + x2 = 4.0069444441

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 4.0069444441

Calculate the square root of the right side: 2.001735358

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 2.001735358 and -2.001735358.

Subproblem 1

x + -0.4166666667 = 2.001735358 Simplifying x + -0.4166666667 = 2.001735358 Reorder the terms: -0.4166666667 + x = 2.001735358 Solving -0.4166666667 + x = 2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 2.001735358 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 2.001735358 + 0.4166666667 x = 2.001735358 + 0.4166666667 Combine like terms: 2.001735358 + 0.4166666667 = 2.4184020247 x = 2.4184020247 Simplifying x = 2.4184020247

Subproblem 2

x + -0.4166666667 = -2.001735358 Simplifying x + -0.4166666667 = -2.001735358 Reorder the terms: -0.4166666667 + x = -2.001735358 Solving -0.4166666667 + x = -2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -2.001735358 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -2.001735358 + 0.4166666667 x = -2.001735358 + 0.4166666667 Combine like terms: -2.001735358 + 0.4166666667 = -1.5850686913 x = -1.5850686913 Simplifying x = -1.5850686913

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.4184020247, -1.5850686913}

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