a^2-9a-54=0

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Solution for a^2-9a-54=0 equation:


Simplifying
a2 + -9a + -54 = 0

Reorder the terms:
-54 + -9a + a2 = 0

Solving
-54 + -9a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -9a + 54 + a2 = 0 + 54

Reorder the terms:
-54 + 54 + -9a + a2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -9a + a2 = 0 + 54
-9a + a2 = 0 + 54

Combine like terms: 0 + 54 = 54
-9a + a2 = 54

The a term is -9a.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9a + 20.25 + a2 = 54 + 20.25

Reorder the terms:
20.25 + -9a + a2 = 54 + 20.25

Combine like terms: 54 + 20.25 = 74.25
20.25 + -9a + a2 = 74.25

Factor a perfect square on the left side:
(a + -4.5)(a + -4.5) = 74.25

Calculate the square root of the right side: 8.61684397

Break this problem into two subproblems by setting 
(a + -4.5) equal to 8.61684397 and -8.61684397.

Subproblem 1

a + -4.5 = 8.61684397 Simplifying a + -4.5 = 8.61684397 Reorder the terms: -4.5 + a = 8.61684397 Solving -4.5 + a = 8.61684397 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = 8.61684397 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = 8.61684397 + 4.5 a = 8.61684397 + 4.5 Combine like terms: 8.61684397 + 4.5 = 13.11684397 a = 13.11684397 Simplifying a = 13.11684397

Subproblem 2

a + -4.5 = -8.61684397 Simplifying a + -4.5 = -8.61684397 Reorder the terms: -4.5 + a = -8.61684397 Solving -4.5 + a = -8.61684397 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = -8.61684397 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = -8.61684397 + 4.5 a = -8.61684397 + 4.5 Combine like terms: -8.61684397 + 4.5 = -4.11684397 a = -4.11684397 Simplifying a = -4.11684397

Solution

The solution to the problem is based on the solutions from the subproblems. a = {13.11684397, -4.11684397}

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