4a^2+a-83=0

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Solution for 4a^2+a-83=0 equation:


Simplifying
4a2 + a + -83 = 0

Reorder the terms:
-83 + a + 4a2 = 0

Solving
-83 + a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
-20.75 + 0.25a + a2 = 0

Move the constant term to the right:

Add '20.75' to each side of the equation.
-20.75 + 0.25a + 20.75 + a2 = 0 + 20.75

Reorder the terms:
-20.75 + 20.75 + 0.25a + a2 = 0 + 20.75

Combine like terms: -20.75 + 20.75 = 0.00
0.00 + 0.25a + a2 = 0 + 20.75
0.25a + a2 = 0 + 20.75

Combine like terms: 0 + 20.75 = 20.75
0.25a + a2 = 20.75

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

Add '0.25' to each side of the equation.
0.25a + 0.25 + a2 = 20.75 + 0.25

Reorder the terms:
0.25 + 0.25a + a2 = 20.75 + 0.25

Combine like terms: 20.75 + 0.25 = 21
0.25 + 0.25a + a2 = 21

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(a + 0.5) equal to 4.582575695 and -4.582575695.

Subproblem 1

a + 0.5 = 4.582575695 Simplifying a + 0.5 = 4.582575695 Reorder the terms: 0.5 + a = 4.582575695 Solving 0.5 + a = 4.582575695 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 4.582575695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.582575695 + -0.5 a = 4.582575695 + -0.5 Combine like terms: 4.582575695 + -0.5 = 4.082575695 a = 4.082575695 Simplifying a = 4.082575695

Subproblem 2

a + 0.5 = -4.582575695 Simplifying a + 0.5 = -4.582575695 Reorder the terms: 0.5 + a = -4.582575695 Solving 0.5 + a = -4.582575695 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -4.582575695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.582575695 + -0.5 a = -4.582575695 + -0.5 Combine like terms: -4.582575695 + -0.5 = -5.082575695 a = -5.082575695 Simplifying a = -5.082575695

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.082575695, -5.082575695}

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