a(29-a)=29

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Solution for a(29-a)=29 equation:


Simplifying
a(29 + -1a) = 29
(29 * a + -1a * a) = 29
(29a + -1a2) = 29

Solving
29a + -1a2 = 29

Solving for variable 'a'.

Reorder the terms:
-29 + 29a + -1a2 = 29 + -29

Combine like terms: 29 + -29 = 0
-29 + 29a + -1a2 = 0

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

Divide each side by '-1'.
29 + -29a + a2 = 0

Move the constant term to the right:

Add '-29' to each side of the equation.
29 + -29a + -29 + a2 = 0 + -29

Reorder the terms:
29 + -29 + -29a + a2 = 0 + -29

Combine like terms: 29 + -29 = 0
0 + -29a + a2 = 0 + -29
-29a + a2 = 0 + -29

Combine like terms: 0 + -29 = -29
-29a + a2 = -29

The a term is -29a.  Take half its coefficient (-14.5).
Square it (210.25) and add it to both sides.

Add '210.25' to each side of the equation.
-29a + 210.25 + a2 = -29 + 210.25

Reorder the terms:
210.25 + -29a + a2 = -29 + 210.25

Combine like terms: -29 + 210.25 = 181.25
210.25 + -29a + a2 = 181.25

Factor a perfect square on the left side:
(a + -14.5)(a + -14.5) = 181.25

Calculate the square root of the right side: 13.462912018

Break this problem into two subproblems by setting 
(a + -14.5) equal to 13.462912018 and -13.462912018.

Subproblem 1

a + -14.5 = 13.462912018 Simplifying a + -14.5 = 13.462912018 Reorder the terms: -14.5 + a = 13.462912018 Solving -14.5 + a = 13.462912018 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + a = 13.462912018 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + a = 13.462912018 + 14.5 a = 13.462912018 + 14.5 Combine like terms: 13.462912018 + 14.5 = 27.962912018 a = 27.962912018 Simplifying a = 27.962912018

Subproblem 2

a + -14.5 = -13.462912018 Simplifying a + -14.5 = -13.462912018 Reorder the terms: -14.5 + a = -13.462912018 Solving -14.5 + a = -13.462912018 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + a = -13.462912018 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + a = -13.462912018 + 14.5 a = -13.462912018 + 14.5 Combine like terms: -13.462912018 + 14.5 = 1.037087982 a = 1.037087982 Simplifying a = 1.037087982

Solution

The solution to the problem is based on the solutions from the subproblems. a = {27.962912018, 1.037087982}

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