6x^2-41x+54=0

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Solution for 6x^2-41x+54=0 equation:


Simplifying
6x2 + -41x + 54 = 0

Reorder the terms:
54 + -41x + 6x2 = 0

Solving
54 + -41x + 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'.
9 + -6.833333333x + x2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -6.833333333x + -9 + x2 = 0 + -9

Reorder the terms:
9 + -9 + -6.833333333x + x2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -6.833333333x + x2 = 0 + -9
-6.833333333x + x2 = 0 + -9

Combine like terms: 0 + -9 = -9
-6.833333333x + x2 = -9

The x term is -6.833333333x.  Take half its coefficient (-3.416666667).
Square it (11.67361111) and add it to both sides.

Add '11.67361111' to each side of the equation.
-6.833333333x + 11.67361111 + x2 = -9 + 11.67361111

Reorder the terms:
11.67361111 + -6.833333333x + x2 = -9 + 11.67361111

Combine like terms: -9 + 11.67361111 = 2.67361111
11.67361111 + -6.833333333x + x2 = 2.67361111

Factor a perfect square on the left side:
(x + -3.416666667)(x + -3.416666667) = 2.67361111

Calculate the square root of the right side: 1.635118072

Break this problem into two subproblems by setting 
(x + -3.416666667) equal to 1.635118072 and -1.635118072.

Subproblem 1

x + -3.416666667 = 1.635118072 Simplifying x + -3.416666667 = 1.635118072 Reorder the terms: -3.416666667 + x = 1.635118072 Solving -3.416666667 + x = 1.635118072 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.416666667' to each side of the equation. -3.416666667 + 3.416666667 + x = 1.635118072 + 3.416666667 Combine like terms: -3.416666667 + 3.416666667 = 0.000000000 0.000000000 + x = 1.635118072 + 3.416666667 x = 1.635118072 + 3.416666667 Combine like terms: 1.635118072 + 3.416666667 = 5.051784739 x = 5.051784739 Simplifying x = 5.051784739

Subproblem 2

x + -3.416666667 = -1.635118072 Simplifying x + -3.416666667 = -1.635118072 Reorder the terms: -3.416666667 + x = -1.635118072 Solving -3.416666667 + x = -1.635118072 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.416666667' to each side of the equation. -3.416666667 + 3.416666667 + x = -1.635118072 + 3.416666667 Combine like terms: -3.416666667 + 3.416666667 = 0.000000000 0.000000000 + x = -1.635118072 + 3.416666667 x = -1.635118072 + 3.416666667 Combine like terms: -1.635118072 + 3.416666667 = 1.781548595 x = 1.781548595 Simplifying x = 1.781548595

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.051784739, 1.781548595}

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