5a(a-1)=35

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Solution for 5a(a-1)=35 equation: 


Simplifying
5a(a + -1) = 35

Reorder the terms:
5a(-1 + a) = 35
(-1 * 5a + a * 5a) = 35
(-5a + 5a2) = 35

Solving
-5a + 5a2 = 35

Solving for variable 'a'.

Reorder the terms:
-35 + -5a + 5a2 = 35 + -35

Combine like terms: 35 + -35 = 0
-35 + -5a + 5a2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-7 + -1a + a2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-7 + -1a + a2)' equal to zero and attempt to solve:

Simplifying
-7 + -1a + a2 = 0

Solving
-7 + -1a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -1a + 7 + a2 = 0 + 7

Reorder the terms:
-7 + 7 + -1a + a2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -1a + a2 = 0 + 7
-1a + a2 = 0 + 7

Combine like terms: 0 + 7 = 7
-1a + a2 = 7

The a term is -1a.  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 = 7 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + -1a + a2 = 7.25

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(a + -0.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
a + -0.5 = 2.692582404

Simplifying
a + -0.5 = 2.692582404

Reorder the terms:
-0.5 + a = 2.692582404

Solving
-0.5 + a = 2.692582404

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 = 2.692582404 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + a = 2.692582404 + 0.5
a = 2.692582404 + 0.5

Combine like terms: 2.692582404 + 0.5 = 3.192582404
a = 3.192582404

Simplifying
a = 3.192582404


Subproblem 2
a + -0.5 = -2.692582404

Simplifying
a + -0.5 = -2.692582404

Reorder the terms:
-0.5 + a = -2.692582404

Solving
-0.5 + a = -2.692582404

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 = -2.692582404 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + a = -2.692582404 + 0.5
a = -2.692582404 + 0.5

Combine like terms: -2.692582404 + 0.5 = -2.192582404
a = -2.192582404

Simplifying
a = -2.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {3.192582404, -2.192582404}
Solution
a = {3.192582404, -2.192582404}

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