6x^2-38x+45=0

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Solution for 6x^2-38x+45=0 equation:


Simplifying
6x2 + -38x + 45 = 0

Reorder the terms:
45 + -38x + 6x2 = 0

Solving
45 + -38x + 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'.
7.5 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '-7.5' to each side of the equation.
7.5 + -6.333333333x + -7.5 + x2 = 0 + -7.5

Reorder the terms:
7.5 + -7.5 + -6.333333333x + x2 = 0 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + -6.333333333x + x2 = 0 + -7.5
-6.333333333x + x2 = 0 + -7.5

Combine like terms: 0 + -7.5 = -7.5
-6.333333333x + x2 = -7.5

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = -7.5 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = -7.5 + 10.02777778

Combine like terms: -7.5 + 10.02777778 = 2.52777778
10.02777778 + -6.333333333x + x2 = 2.52777778

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 2.52777778

Calculate the square root of the right side: 1.58989867

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 1.58989867 and -1.58989867.

Subproblem 1

x + -3.166666667 = 1.58989867 Simplifying x + -3.166666667 = 1.58989867 Reorder the terms: -3.166666667 + x = 1.58989867 Solving -3.166666667 + x = 1.58989867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = 1.58989867 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = 1.58989867 + 3.166666667 x = 1.58989867 + 3.166666667 Combine like terms: 1.58989867 + 3.166666667 = 4.756565337 x = 4.756565337 Simplifying x = 4.756565337

Subproblem 2

x + -3.166666667 = -1.58989867 Simplifying x + -3.166666667 = -1.58989867 Reorder the terms: -3.166666667 + x = -1.58989867 Solving -3.166666667 + x = -1.58989867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = -1.58989867 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = -1.58989867 + 3.166666667 x = -1.58989867 + 3.166666667 Combine like terms: -1.58989867 + 3.166666667 = 1.576767997 x = 1.576767997 Simplifying x = 1.576767997

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.756565337, 1.576767997}

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