0=a^2-9a+12

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Solution for 0=a^2-9a+12 equation:


Simplifying
0 = a2 + -9a + 12

Reorder the terms:
0 = 12 + -9a + a2

Solving
0 = 12 + -9a + a2

Solving for variable 'a'.

Combine like terms: 0 + -12 = -12
-12 + 9a + -1a2 = 12 + -9a + a2 + -12 + 9a + -1a2

Reorder the terms:
-12 + 9a + -1a2 = 12 + -12 + -9a + 9a + a2 + -1a2

Combine like terms: 12 + -12 = 0
-12 + 9a + -1a2 = 0 + -9a + 9a + a2 + -1a2
-12 + 9a + -1a2 = -9a + 9a + a2 + -1a2

Combine like terms: -9a + 9a = 0
-12 + 9a + -1a2 = 0 + a2 + -1a2
-12 + 9a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
-12 + 9a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
12 + -9a + a2 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -9a + -12 + a2 = 0 + -12

Reorder the terms:
12 + -12 + -9a + a2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -9a + a2 = 0 + -12
-9a + a2 = 0 + -12

Combine like terms: 0 + -12 = -12
-9a + a2 = -12

The a term is -9a.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9a + 20.25 + a2 = -12 + 20.25

Reorder the terms:
20.25 + -9a + a2 = -12 + 20.25

Combine like terms: -12 + 20.25 = 8.25
20.25 + -9a + a2 = 8.25

Factor a perfect square on the left side:
(a + -4.5)(a + -4.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(a + -4.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

a + -4.5 = 2.872281323 Simplifying a + -4.5 = 2.872281323 Reorder the terms: -4.5 + a = 2.872281323 Solving -4.5 + a = 2.872281323 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = 2.872281323 + 4.5 a = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 a = 7.372281323 Simplifying a = 7.372281323

Subproblem 2

a + -4.5 = -2.872281323 Simplifying a + -4.5 = -2.872281323 Reorder the terms: -4.5 + a = -2.872281323 Solving -4.5 + a = -2.872281323 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = -2.872281323 + 4.5 a = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 a = 1.627718677 Simplifying a = 1.627718677

Solution

The solution to the problem is based on the solutions from the subproblems. a = {7.372281323, 1.627718677}

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