6x^2+x-52=0

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Solution for 6x^2+x-52=0 equation:


Simplifying
6x2 + x + -52 = 0

Reorder the terms:
-52 + x + 6x2 = 0

Solving
-52 + x + 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'.
-8.666666667 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '8.666666667' to each side of the equation.
-8.666666667 + 0.1666666667x + 8.666666667 + x2 = 0 + 8.666666667

Reorder the terms:
-8.666666667 + 8.666666667 + 0.1666666667x + x2 = 0 + 8.666666667

Combine like terms: -8.666666667 + 8.666666667 = 0.000000000
0.000000000 + 0.1666666667x + x2 = 0 + 8.666666667
0.1666666667x + x2 = 0 + 8.666666667

Combine like terms: 0 + 8.666666667 = 8.666666667
0.1666666667x + x2 = 8.666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667x + 0.25 + x2 = 8.666666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667x + x2 = 8.666666667 + 0.25

Combine like terms: 8.666666667 + 0.25 = 8.916666667
0.25 + 0.1666666667x + x2 = 8.916666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.916666667

Calculate the square root of the right side: 2.986078811

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.986078811 and -2.986078811.

Subproblem 1

x + 0.5 = 2.986078811 Simplifying x + 0.5 = 2.986078811 Reorder the terms: 0.5 + x = 2.986078811 Solving 0.5 + x = 2.986078811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.986078811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.986078811 + -0.5 x = 2.986078811 + -0.5 Combine like terms: 2.986078811 + -0.5 = 2.486078811 x = 2.486078811 Simplifying x = 2.486078811

Subproblem 2

x + 0.5 = -2.986078811 Simplifying x + 0.5 = -2.986078811 Reorder the terms: 0.5 + x = -2.986078811 Solving 0.5 + x = -2.986078811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.986078811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.986078811 + -0.5 x = -2.986078811 + -0.5 Combine like terms: -2.986078811 + -0.5 = -3.486078811 x = -3.486078811 Simplifying x = -3.486078811

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.486078811, -3.486078811}

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