6a^2+19a-15=0

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Solution for 6a^2+19a-15=0 equation:


Simplifying
6a2 + 19a + -15 = 0

Reorder the terms:
-15 + 19a + 6a2 = 0

Solving
-15 + 19a + 6a2 = 0

Solving for variable 'a'.

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

Divide each side by '6'.
-2.5 + 3.166666667a + a2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 3.166666667a + 2.5 + a2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 3.166666667a + a2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 3.166666667a + a2 = 0 + 2.5
3.166666667a + a2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
3.166666667a + a2 = 2.5

The a term is 3.166666667a.  Take half its coefficient (1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
3.166666667a + 2.506944447 + a2 = 2.5 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667a + a2 = 2.5 + 2.506944447

Combine like terms: 2.5 + 2.506944447 = 5.006944447
2.506944447 + 3.166666667a + a2 = 5.006944447

Factor a perfect square on the left side:
(a + 1.583333334)(a + 1.583333334) = 5.006944447

Calculate the square root of the right side: 2.237620264

Break this problem into two subproblems by setting 
(a + 1.583333334) equal to 2.237620264 and -2.237620264.

Subproblem 1

a + 1.583333334 = 2.237620264 Simplifying a + 1.583333334 = 2.237620264 Reorder the terms: 1.583333334 + a = 2.237620264 Solving 1.583333334 + a = 2.237620264 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + a = 2.237620264 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + a = 2.237620264 + -1.583333334 a = 2.237620264 + -1.583333334 Combine like terms: 2.237620264 + -1.583333334 = 0.65428693 a = 0.65428693 Simplifying a = 0.65428693

Subproblem 2

a + 1.583333334 = -2.237620264 Simplifying a + 1.583333334 = -2.237620264 Reorder the terms: 1.583333334 + a = -2.237620264 Solving 1.583333334 + a = -2.237620264 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + a = -2.237620264 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + a = -2.237620264 + -1.583333334 a = -2.237620264 + -1.583333334 Combine like terms: -2.237620264 + -1.583333334 = -3.820953598 a = -3.820953598 Simplifying a = -3.820953598

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.65428693, -3.820953598}

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