3a^2-36a+91=0

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Solution for 3a^2-36a+91=0 equation:


Simplifying
3a2 + -36a + 91 = 0

Reorder the terms:
91 + -36a + 3a2 = 0

Solving
91 + -36a + 3a2 = 0

Solving for variable 'a'.

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

Divide each side by '3'.
30.33333333 + -12a + a2 = 0

Move the constant term to the right:

Add '-30.33333333' to each side of the equation.
30.33333333 + -12a + -30.33333333 + a2 = 0 + -30.33333333

Reorder the terms:
30.33333333 + -30.33333333 + -12a + a2 = 0 + -30.33333333

Combine like terms: 30.33333333 + -30.33333333 = 0.00000000
0.00000000 + -12a + a2 = 0 + -30.33333333
-12a + a2 = 0 + -30.33333333

Combine like terms: 0 + -30.33333333 = -30.33333333
-12a + a2 = -30.33333333

The a term is -12a.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12a + 36 + a2 = -30.33333333 + 36

Reorder the terms:
36 + -12a + a2 = -30.33333333 + 36

Combine like terms: -30.33333333 + 36 = 5.66666667
36 + -12a + a2 = 5.66666667

Factor a perfect square on the left side:
(a + -6)(a + -6) = 5.66666667

Calculate the square root of the right side: 2.380476144

Break this problem into two subproblems by setting 
(a + -6) equal to 2.380476144 and -2.380476144.

Subproblem 1

a + -6 = 2.380476144 Simplifying a + -6 = 2.380476144 Reorder the terms: -6 + a = 2.380476144 Solving -6 + a = 2.380476144 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + a = 2.380476144 + 6 Combine like terms: -6 + 6 = 0 0 + a = 2.380476144 + 6 a = 2.380476144 + 6 Combine like terms: 2.380476144 + 6 = 8.380476144 a = 8.380476144 Simplifying a = 8.380476144

Subproblem 2

a + -6 = -2.380476144 Simplifying a + -6 = -2.380476144 Reorder the terms: -6 + a = -2.380476144 Solving -6 + a = -2.380476144 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + a = -2.380476144 + 6 Combine like terms: -6 + 6 = 0 0 + a = -2.380476144 + 6 a = -2.380476144 + 6 Combine like terms: -2.380476144 + 6 = 3.619523856 a = 3.619523856 Simplifying a = 3.619523856

Solution

The solution to the problem is based on the solutions from the subproblems. a = {8.380476144, 3.619523856}

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