(a^2-4a+6)+(-3a^2+13a+1)=0

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Solution for (a^2-4a+6)+(-3a^2+13a+1)=0 equation:


Simplifying
(a2 + -4a + 6) + (-3a2 + 13a + 1) = 0

Reorder the terms:
(6 + -4a + a2) + (-3a2 + 13a + 1) = 0

Remove parenthesis around (6 + -4a + a2)
6 + -4a + a2 + (-3a2 + 13a + 1) = 0

Reorder the terms:
6 + -4a + a2 + (1 + 13a + -3a2) = 0

Remove parenthesis around (1 + 13a + -3a2)
6 + -4a + a2 + 1 + 13a + -3a2 = 0

Reorder the terms:
6 + 1 + -4a + 13a + a2 + -3a2 = 0

Combine like terms: 6 + 1 = 7
7 + -4a + 13a + a2 + -3a2 = 0

Combine like terms: -4a + 13a = 9a
7 + 9a + a2 + -3a2 = 0

Combine like terms: a2 + -3a2 = -2a2
7 + 9a + -2a2 = 0

Solving
7 + 9a + -2a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-3.5 + -4.5a + a2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -4.5a + 3.5 + a2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -4.5a + a2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -4.5a + a2 = 0 + 3.5
-4.5a + a2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-4.5a + a2 = 3.5

The a term is -4.5a.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5a + 5.0625 + a2 = 3.5 + 5.0625

Reorder the terms:
5.0625 + -4.5a + a2 = 3.5 + 5.0625

Combine like terms: 3.5 + 5.0625 = 8.5625
5.0625 + -4.5a + a2 = 8.5625

Factor a perfect square on the left side:
(a + -2.25)(a + -2.25) = 8.5625

Calculate the square root of the right side: 2.926174978

Break this problem into two subproblems by setting 
(a + -2.25) equal to 2.926174978 and -2.926174978.

Subproblem 1

a + -2.25 = 2.926174978 Simplifying a + -2.25 = 2.926174978 Reorder the terms: -2.25 + a = 2.926174978 Solving -2.25 + a = 2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = 2.926174978 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = 2.926174978 + 2.25 a = 2.926174978 + 2.25 Combine like terms: 2.926174978 + 2.25 = 5.176174978 a = 5.176174978 Simplifying a = 5.176174978

Subproblem 2

a + -2.25 = -2.926174978 Simplifying a + -2.25 = -2.926174978 Reorder the terms: -2.25 + a = -2.926174978 Solving -2.25 + a = -2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = -2.926174978 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = -2.926174978 + 2.25 a = -2.926174978 + 2.25 Combine like terms: -2.926174978 + 2.25 = -0.676174978 a = -0.676174978 Simplifying a = -0.676174978

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.176174978, -0.676174978}

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