12a^2-5a-12=0

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Solution for 12a^2-5a-12=0 equation:


Simplifying
12a2 + -5a + -12 = 0

Reorder the terms:
-12 + -5a + 12a2 = 0

Solving
-12 + -5a + 12a2 = 0

Solving for variable 'a'.

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

Divide each side by '12'.
-1 + -0.4166666667a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.4166666667a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.4166666667a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.4166666667a + a2 = 0 + 1
-0.4166666667a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.4166666667a + a2 = 1

The a term is -0.4166666667a.  Take half its coefficient (-0.2083333334).
Square it (0.04340277781) and add it to both sides.

Add '0.04340277781' to each side of the equation.
-0.4166666667a + 0.04340277781 + a2 = 1 + 0.04340277781

Reorder the terms:
0.04340277781 + -0.4166666667a + a2 = 1 + 0.04340277781

Combine like terms: 1 + 0.04340277781 = 1.04340277781
0.04340277781 + -0.4166666667a + a2 = 1.04340277781

Factor a perfect square on the left side:
(a + -0.2083333334)(a + -0.2083333334) = 1.04340277781

Calculate the square root of the right side: 1.021470889

Break this problem into two subproblems by setting 
(a + -0.2083333334) equal to 1.021470889 and -1.021470889.

Subproblem 1

a + -0.2083333334 = 1.021470889 Simplifying a + -0.2083333334 = 1.021470889 Reorder the terms: -0.2083333334 + a = 1.021470889 Solving -0.2083333334 + a = 1.021470889 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + a = 1.021470889 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + a = 1.021470889 + 0.2083333334 a = 1.021470889 + 0.2083333334 Combine like terms: 1.021470889 + 0.2083333334 = 1.2298042224 a = 1.2298042224 Simplifying a = 1.2298042224

Subproblem 2

a + -0.2083333334 = -1.021470889 Simplifying a + -0.2083333334 = -1.021470889 Reorder the terms: -0.2083333334 + a = -1.021470889 Solving -0.2083333334 + a = -1.021470889 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + a = -1.021470889 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + a = -1.021470889 + 0.2083333334 a = -1.021470889 + 0.2083333334 Combine like terms: -1.021470889 + 0.2083333334 = -0.8131375556 a = -0.8131375556 Simplifying a = -0.8131375556

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.2298042224, -0.8131375556}

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