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

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


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

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

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

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

Combine like terms: -3a + 13a = 10a
6 + -4a + a2 + (1 + 10a) = 0

Remove parenthesis around (1 + 10a)
6 + -4a + a2 + 1 + 10a = 0

Reorder the terms:
6 + 1 + -4a + 10a + a2 = 0

Combine like terms: 6 + 1 = 7
7 + -4a + 10a + a2 = 0

Combine like terms: -4a + 10a = 6a
7 + 6a + a2 = 0

Solving
7 + 6a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 6a + -7 + a2 = 0 + -7

Reorder the terms:
7 + -7 + 6a + a2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 6a + a2 = 0 + -7
6a + a2 = 0 + -7

Combine like terms: 0 + -7 = -7
6a + a2 = -7

The a term is 6a.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6a + 9 + a2 = -7 + 9

Reorder the terms:
9 + 6a + a2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + 6a + a2 = 2

Factor a perfect square on the left side:
(a + 3)(a + 3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(a + 3) equal to 1.414213562 and -1.414213562.

Subproblem 1

a + 3 = 1.414213562 Simplifying a + 3 = 1.414213562 Reorder the terms: 3 + a = 1.414213562 Solving 3 + a = 1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = 1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + a = 1.414213562 + -3 a = 1.414213562 + -3 Combine like terms: 1.414213562 + -3 = -1.585786438 a = -1.585786438 Simplifying a = -1.585786438

Subproblem 2

a + 3 = -1.414213562 Simplifying a + 3 = -1.414213562 Reorder the terms: 3 + a = -1.414213562 Solving 3 + a = -1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = -1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + a = -1.414213562 + -3 a = -1.414213562 + -3 Combine like terms: -1.414213562 + -3 = -4.414213562 a = -4.414213562 Simplifying a = -4.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-1.585786438, -4.414213562}

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