8a^2-4a-31=0

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Solution for 8a^2-4a-31=0 equation:


Simplifying
8a2 + -4a + -31 = 0

Reorder the terms:
-31 + -4a + 8a2 = 0

Solving
-31 + -4a + 8a2 = 0

Solving for variable 'a'.

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

Divide each side by '8'.
-3.875 + -0.5a + a2 = 0

Move the constant term to the right:

Add '3.875' to each side of the equation.
-3.875 + -0.5a + 3.875 + a2 = 0 + 3.875

Reorder the terms:
-3.875 + 3.875 + -0.5a + a2 = 0 + 3.875

Combine like terms: -3.875 + 3.875 = 0.000
0.000 + -0.5a + a2 = 0 + 3.875
-0.5a + a2 = 0 + 3.875

Combine like terms: 0 + 3.875 = 3.875
-0.5a + a2 = 3.875

The a term is -0.5a.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5a + 0.0625 + a2 = 3.875 + 0.0625

Reorder the terms:
0.0625 + -0.5a + a2 = 3.875 + 0.0625

Combine like terms: 3.875 + 0.0625 = 3.9375
0.0625 + -0.5a + a2 = 3.9375

Factor a perfect square on the left side:
(a + -0.25)(a + -0.25) = 3.9375

Calculate the square root of the right side: 1.984313483

Break this problem into two subproblems by setting 
(a + -0.25) equal to 1.984313483 and -1.984313483.

Subproblem 1

a + -0.25 = 1.984313483 Simplifying a + -0.25 = 1.984313483 Reorder the terms: -0.25 + a = 1.984313483 Solving -0.25 + a = 1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.984313483 + 0.25 a = 1.984313483 + 0.25 Combine like terms: 1.984313483 + 0.25 = 2.234313483 a = 2.234313483 Simplifying a = 2.234313483

Subproblem 2

a + -0.25 = -1.984313483 Simplifying a + -0.25 = -1.984313483 Reorder the terms: -0.25 + a = -1.984313483 Solving -0.25 + a = -1.984313483 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -1.984313483 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.984313483 + 0.25 a = -1.984313483 + 0.25 Combine like terms: -1.984313483 + 0.25 = -1.734313483 a = -1.734313483 Simplifying a = -1.734313483

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.234313483, -1.734313483}

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