13a^2-78a-108=0

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Solution for 13a^2-78a-108=0 equation:


Simplifying
13a2 + -78a + -108 = 0

Reorder the terms:
-108 + -78a + 13a2 = 0

Solving
-108 + -78a + 13a2 = 0

Solving for variable 'a'.

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

Divide each side by '13'.
-8.307692308 + -6a + a2 = 0

Move the constant term to the right:

Add '8.307692308' to each side of the equation.
-8.307692308 + -6a + 8.307692308 + a2 = 0 + 8.307692308

Reorder the terms:
-8.307692308 + 8.307692308 + -6a + a2 = 0 + 8.307692308

Combine like terms: -8.307692308 + 8.307692308 = 0.000000000
0.000000000 + -6a + a2 = 0 + 8.307692308
-6a + a2 = 0 + 8.307692308

Combine like terms: 0 + 8.307692308 = 8.307692308
-6a + a2 = 8.307692308

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 = 8.307692308 + 9

Reorder the terms:
9 + -6a + a2 = 8.307692308 + 9

Combine like terms: 8.307692308 + 9 = 17.307692308
9 + -6a + a2 = 17.307692308

Factor a perfect square on the left side:
(a + -3)(a + -3) = 17.307692308

Calculate the square root of the right side: 4.160251472

Break this problem into two subproblems by setting 
(a + -3) equal to 4.160251472 and -4.160251472.

Subproblem 1

a + -3 = 4.160251472 Simplifying a + -3 = 4.160251472 Reorder the terms: -3 + a = 4.160251472 Solving -3 + a = 4.160251472 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 = 4.160251472 + 3 Combine like terms: -3 + 3 = 0 0 + a = 4.160251472 + 3 a = 4.160251472 + 3 Combine like terms: 4.160251472 + 3 = 7.160251472 a = 7.160251472 Simplifying a = 7.160251472

Subproblem 2

a + -3 = -4.160251472 Simplifying a + -3 = -4.160251472 Reorder the terms: -3 + a = -4.160251472 Solving -3 + a = -4.160251472 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 = -4.160251472 + 3 Combine like terms: -3 + 3 = 0 0 + a = -4.160251472 + 3 a = -4.160251472 + 3 Combine like terms: -4.160251472 + 3 = -1.160251472 a = -1.160251472 Simplifying a = -1.160251472

Solution

The solution to the problem is based on the solutions from the subproblems. a = {7.160251472, -1.160251472}

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