2a+6-a^2=3-3a

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Solution for 2a+6-a^2=3-3a equation:


Simplifying
2a + 6 + -1a2 = 3 + -3a

Reorder the terms:
6 + 2a + -1a2 = 3 + -3a

Solving
6 + 2a + -1a2 = 3 + -3a

Solving for variable 'a'.

Reorder the terms:
6 + -3 + 2a + 3a + -1a2 = 3 + -3a + -3 + 3a

Combine like terms: 6 + -3 = 3
3 + 2a + 3a + -1a2 = 3 + -3a + -3 + 3a

Combine like terms: 2a + 3a = 5a
3 + 5a + -1a2 = 3 + -3a + -3 + 3a

Reorder the terms:
3 + 5a + -1a2 = 3 + -3 + -3a + 3a

Combine like terms: 3 + -3 = 0
3 + 5a + -1a2 = 0 + -3a + 3a
3 + 5a + -1a2 = -3a + 3a

Combine like terms: -3a + 3a = 0
3 + 5a + -1a2 = 0

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

Divide each side by '-1'.
-3 + -5a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -5a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + -5a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -5a + a2 = 0 + 3
-5a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-5a + a2 = 3

The a term is -5a.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5a + 6.25 + a2 = 3 + 6.25

Reorder the terms:
6.25 + -5a + a2 = 3 + 6.25

Combine like terms: 3 + 6.25 = 9.25
6.25 + -5a + a2 = 9.25

Factor a perfect square on the left side:
(a + -2.5)(a + -2.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(a + -2.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

a + -2.5 = 3.041381265 Simplifying a + -2.5 = 3.041381265 Reorder the terms: -2.5 + a = 3.041381265 Solving -2.5 + a = 3.041381265 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 3.041381265 + 2.5 a = 3.041381265 + 2.5 Combine like terms: 3.041381265 + 2.5 = 5.541381265 a = 5.541381265 Simplifying a = 5.541381265

Subproblem 2

a + -2.5 = -3.041381265 Simplifying a + -2.5 = -3.041381265 Reorder the terms: -2.5 + a = -3.041381265 Solving -2.5 + a = -3.041381265 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -3.041381265 + 2.5 a = -3.041381265 + 2.5 Combine like terms: -3.041381265 + 2.5 = -0.541381265 a = -0.541381265 Simplifying a = -0.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.541381265, -0.541381265}

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