15a^2+7a-3=0

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Solution for 15a^2+7a-3=0 equation: 


Simplifying
15a2 + 7a + -3 = 0

Reorder the terms:
-3 + 7a + 15a2 = 0

Solving
-3 + 7a + 15a2 = 0

Solving for variable 'a'.

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

Divide each side by '15'.
-0.2 + 0.4666666667a + a2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 0.4666666667a + 0.2 + a2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 0.4666666667a + a2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 0.4666666667a + a2 = 0 + 0.2
0.4666666667a + a2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
0.4666666667a + a2 = 0.2

The a term is 0.4666666667a.  Take half its coefficient (0.2333333334).
Square it (0.05444444448) and add it to both sides.

Add '0.05444444448' to each side of the equation.
0.4666666667a + 0.05444444448 + a2 = 0.2 + 0.05444444448

Reorder the terms:
0.05444444448 + 0.4666666667a + a2 = 0.2 + 0.05444444448

Combine like terms: 0.2 + 0.05444444448 = 0.25444444448
0.05444444448 + 0.4666666667a + a2 = 0.25444444448

Factor a perfect square on the left side:
(a + 0.2333333334)(a + 0.2333333334) = 0.25444444448

Calculate the square root of the right side: 0.504424865

Break this problem into two subproblems by setting 
(a + 0.2333333334) equal to 0.504424865 and -0.504424865.


Subproblem 1
a + 0.2333333334 = 0.504424865

Simplifying
a + 0.2333333334 = 0.504424865

Reorder the terms:
0.2333333334 + a = 0.504424865

Solving
0.2333333334 + a = 0.504424865

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + a = 0.504424865 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + a = 0.504424865 + -0.2333333334
a = 0.504424865 + -0.2333333334

Combine like terms: 0.504424865 + -0.2333333334 = 0.2710915316
a = 0.2710915316

Simplifying
a = 0.2710915316


Subproblem 2
a + 0.2333333334 = -0.504424865

Simplifying
a + 0.2333333334 = -0.504424865

Reorder the terms:
0.2333333334 + a = -0.504424865

Solving
0.2333333334 + a = -0.504424865

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + a = -0.504424865 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + a = -0.504424865 + -0.2333333334
a = -0.504424865 + -0.2333333334

Combine like terms: -0.504424865 + -0.2333333334 = -0.7377581984
a = -0.7377581984

Simplifying
a = -0.7377581984


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.2710915316, -0.7377581984}

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