a^2-a-19=0

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Solution for a^2-a-19=0 equation:


Simplifying
a2 + -1a + -19 = 0

Reorder the terms:
-19 + -1a + a2 = 0

Solving
-19 + -1a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -1a + 19 + a2 = 0 + 19

Reorder the terms:
-19 + 19 + -1a + a2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -1a + a2 = 0 + 19
-1a + a2 = 0 + 19

Combine like terms: 0 + 19 = 19
-1a + a2 = 19

The a term is -1a.  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.
-1a + 0.25 + a2 = 19 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 19 + 0.25

Combine like terms: 19 + 0.25 = 19.25
0.25 + -1a + a2 = 19.25

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(a + -0.5) equal to 4.387482194 and -4.387482194.

Subproblem 1

a + -0.5 = 4.387482194 Simplifying a + -0.5 = 4.387482194 Reorder the terms: -0.5 + a = 4.387482194 Solving -0.5 + a = 4.387482194 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 = 4.387482194 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 4.387482194 + 0.5 a = 4.387482194 + 0.5 Combine like terms: 4.387482194 + 0.5 = 4.887482194 a = 4.887482194 Simplifying a = 4.887482194

Subproblem 2

a + -0.5 = -4.387482194 Simplifying a + -0.5 = -4.387482194 Reorder the terms: -0.5 + a = -4.387482194 Solving -0.5 + a = -4.387482194 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 = -4.387482194 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -4.387482194 + 0.5 a = -4.387482194 + 0.5 Combine like terms: -4.387482194 + 0.5 = -3.887482194 a = -3.887482194 Simplifying a = -3.887482194

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.887482194, -3.887482194}

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