a=a(a-21)-100

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Solution for a=a(a-21)-100 equation:


Simplifying
a = a(a + -21) + -100

Reorder the terms:
a = a(-21 + a) + -100
a = (-21 * a + a * a) + -100
a = (-21a + a2) + -100

Reorder the terms:
a = -100 + -21a + a2

Solving
a = -100 + -21a + a2

Solving for variable 'a'.

Reorder the terms:
100 + a + 21a + -1a2 = -100 + -21a + a2 + 100 + 21a + -1a2

Combine like terms: a + 21a = 22a
100 + 22a + -1a2 = -100 + -21a + a2 + 100 + 21a + -1a2

Reorder the terms:
100 + 22a + -1a2 = -100 + 100 + -21a + 21a + a2 + -1a2

Combine like terms: -100 + 100 = 0
100 + 22a + -1a2 = 0 + -21a + 21a + a2 + -1a2
100 + 22a + -1a2 = -21a + 21a + a2 + -1a2

Combine like terms: -21a + 21a = 0
100 + 22a + -1a2 = 0 + a2 + -1a2
100 + 22a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
100 + 22a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-100 + -22a + a2 = 0

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + -22a + 100 + a2 = 0 + 100

Reorder the terms:
-100 + 100 + -22a + a2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + -22a + a2 = 0 + 100
-22a + a2 = 0 + 100

Combine like terms: 0 + 100 = 100
-22a + a2 = 100

The a term is -22a.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22a + 121 + a2 = 100 + 121

Reorder the terms:
121 + -22a + a2 = 100 + 121

Combine like terms: 100 + 121 = 221
121 + -22a + a2 = 221

Factor a perfect square on the left side:
(a + -11)(a + -11) = 221

Calculate the square root of the right side: 14.866068747

Break this problem into two subproblems by setting 
(a + -11) equal to 14.866068747 and -14.866068747.

Subproblem 1

a + -11 = 14.866068747 Simplifying a + -11 = 14.866068747 Reorder the terms: -11 + a = 14.866068747 Solving -11 + a = 14.866068747 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + a = 14.866068747 + 11 Combine like terms: -11 + 11 = 0 0 + a = 14.866068747 + 11 a = 14.866068747 + 11 Combine like terms: 14.866068747 + 11 = 25.866068747 a = 25.866068747 Simplifying a = 25.866068747

Subproblem 2

a + -11 = -14.866068747 Simplifying a + -11 = -14.866068747 Reorder the terms: -11 + a = -14.866068747 Solving -11 + a = -14.866068747 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + a = -14.866068747 + 11 Combine like terms: -11 + 11 = 0 0 + a = -14.866068747 + 11 a = -14.866068747 + 11 Combine like terms: -14.866068747 + 11 = -3.866068747 a = -3.866068747 Simplifying a = -3.866068747

Solution

The solution to the problem is based on the solutions from the subproblems. a = {25.866068747, -3.866068747}

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