(a-9)(a-19)=a

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Solution for (a-9)(a-19)=a equation: 


Simplifying
(a + -9)(a + -19) = a

Reorder the terms:
(-9 + a)(a + -19) = a

Reorder the terms:
(-9 + a)(-19 + a) = a

Multiply (-9 + a) * (-19 + a)
(-9(-19 + a) + a(-19 + a)) = a
((-19 * -9 + a * -9) + a(-19 + a)) = a
((171 + -9a) + a(-19 + a)) = a
(171 + -9a + (-19 * a + a * a)) = a
(171 + -9a + (-19a + a2)) = a

Combine like terms: -9a + -19a = -28a
(171 + -28a + a2) = a

Solving
171 + -28a + a2 = a

Solving for variable 'a'.

Reorder the terms:
171 + -28a + -1a + a2 = a + -1a

Combine like terms: -28a + -1a = -29a
171 + -29a + a2 = a + -1a

Combine like terms: a + -1a = 0
171 + -29a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-171' to each side of the equation.
171 + -29a + -171 + a2 = 0 + -171

Reorder the terms:
171 + -171 + -29a + a2 = 0 + -171

Combine like terms: 171 + -171 = 0
0 + -29a + a2 = 0 + -171
-29a + a2 = 0 + -171

Combine like terms: 0 + -171 = -171
-29a + a2 = -171

The a term is -29a.  Take half its coefficient (-14.5).
Square it (210.25) and add it to both sides.

Add '210.25' to each side of the equation.
-29a + 210.25 + a2 = -171 + 210.25

Reorder the terms:
210.25 + -29a + a2 = -171 + 210.25

Combine like terms: -171 + 210.25 = 39.25
210.25 + -29a + a2 = 39.25

Factor a perfect square on the left side:
(a + -14.5)(a + -14.5) = 39.25

Calculate the square root of the right side: 6.264982043

Break this problem into two subproblems by setting 
(a + -14.5) equal to 6.264982043 and -6.264982043.


Subproblem 1
a + -14.5 = 6.264982043

Simplifying
a + -14.5 = 6.264982043

Reorder the terms:
-14.5 + a = 6.264982043

Solving
-14.5 + a = 6.264982043

Solving for variable 'a'.

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

Add '14.5' to each side of the equation.
-14.5 + 14.5 + a = 6.264982043 + 14.5

Combine like terms: -14.5 + 14.5 = 0.0
0.0 + a = 6.264982043 + 14.5
a = 6.264982043 + 14.5

Combine like terms: 6.264982043 + 14.5 = 20.764982043
a = 20.764982043

Simplifying
a = 20.764982043


Subproblem 2
a + -14.5 = -6.264982043

Simplifying
a + -14.5 = -6.264982043

Reorder the terms:
-14.5 + a = -6.264982043

Solving
-14.5 + a = -6.264982043

Solving for variable 'a'.

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

Add '14.5' to each side of the equation.
-14.5 + 14.5 + a = -6.264982043 + 14.5

Combine like terms: -14.5 + 14.5 = 0.0
0.0 + a = -6.264982043 + 14.5
a = -6.264982043 + 14.5

Combine like terms: -6.264982043 + 14.5 = 8.235017957
a = 8.235017957

Simplifying
a = 8.235017957


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {20.764982043, 8.235017957}

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