a^2-21a-108=0

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Solution for a^2-21a-108=0 equation:


Simplifying
a2 + -21a + -108 = 0

Reorder the terms:
-108 + -21a + a2 = 0

Solving
-108 + -21a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + -21a + 108 + a2 = 0 + 108

Reorder the terms:
-108 + 108 + -21a + a2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + -21a + a2 = 0 + 108
-21a + a2 = 0 + 108

Combine like terms: 0 + 108 = 108
-21a + a2 = 108

The a term is -21a.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21a + 110.25 + a2 = 108 + 110.25

Reorder the terms:
110.25 + -21a + a2 = 108 + 110.25

Combine like terms: 108 + 110.25 = 218.25
110.25 + -21a + a2 = 218.25

Factor a perfect square on the left side:
(a + -10.5)(a + -10.5) = 218.25

Calculate the square root of the right side: 14.773286703

Break this problem into two subproblems by setting 
(a + -10.5) equal to 14.773286703 and -14.773286703.

Subproblem 1

a + -10.5 = 14.773286703 Simplifying a + -10.5 = 14.773286703 Reorder the terms: -10.5 + a = 14.773286703 Solving -10.5 + a = 14.773286703 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + a = 14.773286703 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + a = 14.773286703 + 10.5 a = 14.773286703 + 10.5 Combine like terms: 14.773286703 + 10.5 = 25.273286703 a = 25.273286703 Simplifying a = 25.273286703

Subproblem 2

a + -10.5 = -14.773286703 Simplifying a + -10.5 = -14.773286703 Reorder the terms: -10.5 + a = -14.773286703 Solving -10.5 + a = -14.773286703 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + a = -14.773286703 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + a = -14.773286703 + 10.5 a = -14.773286703 + 10.5 Combine like terms: -14.773286703 + 10.5 = -4.273286703 a = -4.273286703 Simplifying a = -4.273286703

Solution

The solution to the problem is based on the solutions from the subproblems. a = {25.273286703, -4.273286703}

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