12x^2-65x+72=0

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Solution for 12x^2-65x+72=0 equation:


Simplifying
12x2 + -65x + 72 = 0

Reorder the terms:
72 + -65x + 12x2 = 0

Solving
72 + -65x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
6 + -5.416666667x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -5.416666667x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + -5.416666667x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -5.416666667x + x2 = 0 + -6
-5.416666667x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
-5.416666667x + x2 = -6

The x term is -5.416666667x.  Take half its coefficient (-2.708333334).
Square it (7.335069448) and add it to both sides.

Add '7.335069448' to each side of the equation.
-5.416666667x + 7.335069448 + x2 = -6 + 7.335069448

Reorder the terms:
7.335069448 + -5.416666667x + x2 = -6 + 7.335069448

Combine like terms: -6 + 7.335069448 = 1.335069448
7.335069448 + -5.416666667x + x2 = 1.335069448

Factor a perfect square on the left side:
(x + -2.708333334)(x + -2.708333334) = 1.335069448

Calculate the square root of the right side: 1.155452054

Break this problem into two subproblems by setting 
(x + -2.708333334) equal to 1.155452054 and -1.155452054.

Subproblem 1

x + -2.708333334 = 1.155452054 Simplifying x + -2.708333334 = 1.155452054 Reorder the terms: -2.708333334 + x = 1.155452054 Solving -2.708333334 + x = 1.155452054 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.708333334' to each side of the equation. -2.708333334 + 2.708333334 + x = 1.155452054 + 2.708333334 Combine like terms: -2.708333334 + 2.708333334 = 0.000000000 0.000000000 + x = 1.155452054 + 2.708333334 x = 1.155452054 + 2.708333334 Combine like terms: 1.155452054 + 2.708333334 = 3.863785388 x = 3.863785388 Simplifying x = 3.863785388

Subproblem 2

x + -2.708333334 = -1.155452054 Simplifying x + -2.708333334 = -1.155452054 Reorder the terms: -2.708333334 + x = -1.155452054 Solving -2.708333334 + x = -1.155452054 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.708333334' to each side of the equation. -2.708333334 + 2.708333334 + x = -1.155452054 + 2.708333334 Combine like terms: -2.708333334 + 2.708333334 = 0.000000000 0.000000000 + x = -1.155452054 + 2.708333334 x = -1.155452054 + 2.708333334 Combine like terms: -1.155452054 + 2.708333334 = 1.55288128 x = 1.55288128 Simplifying x = 1.55288128

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.863785388, 1.55288128}

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