6a^2=-5a+13

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Solution for 6a^2=-5a+13 equation:


Simplifying
6a2 = -5a + 13

Reorder the terms:
6a2 = 13 + -5a

Solving
6a2 = 13 + -5a

Solving for variable 'a'.

Reorder the terms:
-13 + 5a + 6a2 = 13 + -5a + -13 + 5a

Reorder the terms:
-13 + 5a + 6a2 = 13 + -13 + -5a + 5a

Combine like terms: 13 + -13 = 0
-13 + 5a + 6a2 = 0 + -5a + 5a
-13 + 5a + 6a2 = -5a + 5a

Combine like terms: -5a + 5a = 0
-13 + 5a + 6a2 = 0

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

Divide each side by '6'.
-2.166666667 + 0.8333333333a + a2 = 0

Move the constant term to the right:

Add '2.166666667' to each side of the equation.
-2.166666667 + 0.8333333333a + 2.166666667 + a2 = 0 + 2.166666667

Reorder the terms:
-2.166666667 + 2.166666667 + 0.8333333333a + a2 = 0 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + 0.8333333333a + a2 = 0 + 2.166666667
0.8333333333a + a2 = 0 + 2.166666667

Combine like terms: 0 + 2.166666667 = 2.166666667
0.8333333333a + a2 = 2.166666667

The a term is 0.8333333333a.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333a + 0.1736111111 + a2 = 2.166666667 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333a + a2 = 2.166666667 + 0.1736111111

Combine like terms: 2.166666667 + 0.1736111111 = 2.3402777781
0.1736111111 + 0.8333333333a + a2 = 2.3402777781

Factor a perfect square on the left side:
(a + 0.4166666667)(a + 0.4166666667) = 2.3402777781

Calculate the square root of the right side: 1.529796646

Break this problem into two subproblems by setting 
(a + 0.4166666667) equal to 1.529796646 and -1.529796646.

Subproblem 1

a + 0.4166666667 = 1.529796646 Simplifying a + 0.4166666667 = 1.529796646 Reorder the terms: 0.4166666667 + a = 1.529796646 Solving 0.4166666667 + a = 1.529796646 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + a = 1.529796646 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + a = 1.529796646 + -0.4166666667 a = 1.529796646 + -0.4166666667 Combine like terms: 1.529796646 + -0.4166666667 = 1.1131299793 a = 1.1131299793 Simplifying a = 1.1131299793

Subproblem 2

a + 0.4166666667 = -1.529796646 Simplifying a + 0.4166666667 = -1.529796646 Reorder the terms: 0.4166666667 + a = -1.529796646 Solving 0.4166666667 + a = -1.529796646 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + a = -1.529796646 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + a = -1.529796646 + -0.4166666667 a = -1.529796646 + -0.4166666667 Combine like terms: -1.529796646 + -0.4166666667 = -1.9464633127 a = -1.9464633127 Simplifying a = -1.9464633127

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.1131299793, -1.9464633127}

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