15a^2-16a-3=0

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Solution for 15a^2-16a-3=0 equation:


Simplifying
15a2 + -16a + -3 = 0

Reorder the terms:
-3 + -16a + 15a2 = 0

Solving
-3 + -16a + 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 + -1.066666667a + a2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -1.066666667a + 0.2 + a2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -1.066666667a + a2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -1.066666667a + a2 = 0 + 0.2
-1.066666667a + a2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-1.066666667a + a2 = 0.2

The a term is -1.066666667a.  Take half its coefficient (-0.5333333335).
Square it (0.2844444446) and add it to both sides.

Add '0.2844444446' to each side of the equation.
-1.066666667a + 0.2844444446 + a2 = 0.2 + 0.2844444446

Reorder the terms:
0.2844444446 + -1.066666667a + a2 = 0.2 + 0.2844444446

Combine like terms: 0.2 + 0.2844444446 = 0.4844444446
0.2844444446 + -1.066666667a + a2 = 0.4844444446

Factor a perfect square on the left side:
(a + -0.5333333335)(a + -0.5333333335) = 0.4844444446

Calculate the square root of the right side: 0.696020434

Break this problem into two subproblems by setting 
(a + -0.5333333335) equal to 0.696020434 and -0.696020434.

Subproblem 1

a + -0.5333333335 = 0.696020434 Simplifying a + -0.5333333335 = 0.696020434 Reorder the terms: -0.5333333335 + a = 0.696020434 Solving -0.5333333335 + a = 0.696020434 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5333333335' to each side of the equation. -0.5333333335 + 0.5333333335 + a = 0.696020434 + 0.5333333335 Combine like terms: -0.5333333335 + 0.5333333335 = 0.0000000000 0.0000000000 + a = 0.696020434 + 0.5333333335 a = 0.696020434 + 0.5333333335 Combine like terms: 0.696020434 + 0.5333333335 = 1.2293537675 a = 1.2293537675 Simplifying a = 1.2293537675

Subproblem 2

a + -0.5333333335 = -0.696020434 Simplifying a + -0.5333333335 = -0.696020434 Reorder the terms: -0.5333333335 + a = -0.696020434 Solving -0.5333333335 + a = -0.696020434 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5333333335' to each side of the equation. -0.5333333335 + 0.5333333335 + a = -0.696020434 + 0.5333333335 Combine like terms: -0.5333333335 + 0.5333333335 = 0.0000000000 0.0000000000 + a = -0.696020434 + 0.5333333335 a = -0.696020434 + 0.5333333335 Combine like terms: -0.696020434 + 0.5333333335 = -0.1626871005 a = -0.1626871005 Simplifying a = -0.1626871005

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.2293537675, -0.1626871005}

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