6x^2-46x+45=0

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Solution for 6x^2-46x+45=0 equation: 


Simplifying
6x2 + -46x + 45 = 0

Reorder the terms:
45 + -46x + 6x2 = 0

Solving
45 + -46x + 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 + -7.666666667x + x2 = 0

Move the constant term to the right:

Add '-7.5' to each side of the equation.
7.5 + -7.666666667x + -7.5 + x2 = 0 + -7.5

Reorder the terms:
7.5 + -7.5 + -7.666666667x + x2 = 0 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + -7.666666667x + x2 = 0 + -7.5
-7.666666667x + x2 = 0 + -7.5

Combine like terms: 0 + -7.5 = -7.5
-7.666666667x + x2 = -7.5

The x term is -7.666666667x.  Take half its coefficient (-3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
-7.666666667x + 14.69444445 + x2 = -7.5 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667x + x2 = -7.5 + 14.69444445

Combine like terms: -7.5 + 14.69444445 = 7.19444445
14.69444445 + -7.666666667x + x2 = 7.19444445

Factor a perfect square on the left side:
(x + -3.833333334)(x + -3.833333334) = 7.19444445

Calculate the square root of the right side: 2.682246158

Break this problem into two subproblems by setting 
(x + -3.833333334) equal to 2.682246158 and -2.682246158.


Subproblem 1
x + -3.833333334 = 2.682246158

Simplifying
x + -3.833333334 = 2.682246158

Reorder the terms:
-3.833333334 + x = 2.682246158

Solving
-3.833333334 + x = 2.682246158

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + x = 2.682246158 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = 2.682246158 + 3.833333334
x = 2.682246158 + 3.833333334

Combine like terms: 2.682246158 + 3.833333334 = 6.515579492
x = 6.515579492

Simplifying
x = 6.515579492


Subproblem 2
x + -3.833333334 = -2.682246158

Simplifying
x + -3.833333334 = -2.682246158

Reorder the terms:
-3.833333334 + x = -2.682246158

Solving
-3.833333334 + x = -2.682246158

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + x = -2.682246158 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = -2.682246158 + 3.833333334
x = -2.682246158 + 3.833333334

Combine like terms: -2.682246158 + 3.833333334 = 1.151087176
x = 1.151087176

Simplifying
x = 1.151087176


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.515579492, 1.151087176}

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