18=a(a-1)

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Solution for 18=a(a-1) equation:


Simplifying
18 = a(a + -1)

Reorder the terms:
18 = a(-1 + a)
18 = (-1 * a + a * a)
18 = (-1a + a2)

Solving
18 = -1a + a2

Solving for variable 'a'.

Reorder the terms:
18 + a + -1a2 = -1a + a + a2 + -1a2

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

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

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

Divide each side by '-1'.
-18 + -1a + a2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + -1a + 18 + a2 = 0 + 18

Reorder the terms:
-18 + 18 + -1a + a2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + -1a + a2 = 0 + 18
-1a + a2 = 0 + 18

Combine like terms: 0 + 18 = 18
-1a + a2 = 18

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.
-1a + 0.25 + a2 = 18 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 18 + 0.25

Combine like terms: 18 + 0.25 = 18.25
0.25 + -1a + a2 = 18.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(a + 0.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

a + 0.5 = 4.272001873 Simplifying a + 0.5 = 4.272001873 Reorder the terms: 0.5 + a = 4.272001873 Solving 0.5 + a = 4.272001873 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 = 4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.272001873 + -0.5 a = 4.272001873 + -0.5 Combine like terms: 4.272001873 + -0.5 = 3.772001873 a = 3.772001873 Simplifying a = 3.772001873

Subproblem 2

a + 0.5 = -4.272001873 Simplifying a + 0.5 = -4.272001873 Reorder the terms: 0.5 + a = -4.272001873 Solving 0.5 + a = -4.272001873 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 = -4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.272001873 + -0.5 a = -4.272001873 + -0.5 Combine like terms: -4.272001873 + -0.5 = -4.772001873 a = -4.772001873 Simplifying a = -4.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.772001873, -4.772001873}

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