(a*a)-21a-65=0

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Solution for (a*a)-21a-65=0 equation:


Simplifying
(a * a) + -21a + -65 = 0

Multiply a * a
(a2) + -21a + -65 = 0
a2 + -21a + -65 = 0

Reorder the terms:
-65 + -21a + a2 = 0

Solving
-65 + -21a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '65' to each side of the equation.
-65 + -21a + 65 + a2 = 0 + 65

Reorder the terms:
-65 + 65 + -21a + a2 = 0 + 65

Combine like terms: -65 + 65 = 0
0 + -21a + a2 = 0 + 65
-21a + a2 = 0 + 65

Combine like terms: 0 + 65 = 65
-21a + a2 = 65

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 = 65 + 110.25

Reorder the terms:
110.25 + -21a + a2 = 65 + 110.25

Combine like terms: 65 + 110.25 = 175.25
110.25 + -21a + a2 = 175.25

Factor a perfect square on the left side:
(a + -10.5)(a + -10.5) = 175.25

Calculate the square root of the right side: 13.238202295

Break this problem into two subproblems by setting 
(a + -10.5) equal to 13.238202295 and -13.238202295.

Subproblem 1

a + -10.5 = 13.238202295 Simplifying a + -10.5 = 13.238202295 Reorder the terms: -10.5 + a = 13.238202295 Solving -10.5 + a = 13.238202295 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 = 13.238202295 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + a = 13.238202295 + 10.5 a = 13.238202295 + 10.5 Combine like terms: 13.238202295 + 10.5 = 23.738202295 a = 23.738202295 Simplifying a = 23.738202295

Subproblem 2

a + -10.5 = -13.238202295 Simplifying a + -10.5 = -13.238202295 Reorder the terms: -10.5 + a = -13.238202295 Solving -10.5 + a = -13.238202295 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 = -13.238202295 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + a = -13.238202295 + 10.5 a = -13.238202295 + 10.5 Combine like terms: -13.238202295 + 10.5 = -2.738202295 a = -2.738202295 Simplifying a = -2.738202295

Solution

The solution to the problem is based on the solutions from the subproblems. a = {23.738202295, -2.738202295}

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