5a(a-2)-3a^2=20

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Solution for 5a(a-2)-3a^2=20 equation: 


Simplifying
5a(a + -2) + -3a2 = 20

Reorder the terms:
5a(-2 + a) + -3a2 = 20
(-2 * 5a + a * 5a) + -3a2 = 20
(-10a + 5a2) + -3a2 = 20

Combine like terms: 5a2 + -3a2 = 2a2
-10a + 2a2 = 20

Solving
-10a + 2a2 = 20

Solving for variable 'a'.

Reorder the terms:
-20 + -10a + 2a2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + -10a + 2a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-10 + -5a + a2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-10 + -5a + a2)' equal to zero and attempt to solve:

Simplifying
-10 + -5a + a2 = 0

Solving
-10 + -5a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -5a + 10 + a2 = 0 + 10

Reorder the terms:
-10 + 10 + -5a + a2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -5a + a2 = 0 + 10
-5a + a2 = 0 + 10

Combine like terms: 0 + 10 = 10
-5a + a2 = 10

The a term is -5a.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5a + 6.25 + a2 = 10 + 6.25

Reorder the terms:
6.25 + -5a + a2 = 10 + 6.25

Combine like terms: 10 + 6.25 = 16.25
6.25 + -5a + a2 = 16.25

Factor a perfect square on the left side:
(a + -2.5)(a + -2.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(a + -2.5) equal to 4.031128874 and -4.031128874.


Subproblem 1
a + -2.5 = 4.031128874

Simplifying
a + -2.5 = 4.031128874

Reorder the terms:
-2.5 + a = 4.031128874

Solving
-2.5 + a = 4.031128874

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + a = 4.031128874 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + a = 4.031128874 + 2.5
a = 4.031128874 + 2.5

Combine like terms: 4.031128874 + 2.5 = 6.531128874
a = 6.531128874

Simplifying
a = 6.531128874


Subproblem 2
a + -2.5 = -4.031128874

Simplifying
a + -2.5 = -4.031128874

Reorder the terms:
-2.5 + a = -4.031128874

Solving
-2.5 + a = -4.031128874

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + a = -4.031128874 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + a = -4.031128874 + 2.5
a = -4.031128874 + 2.5

Combine like terms: -4.031128874 + 2.5 = -1.531128874
a = -1.531128874

Simplifying
a = -1.531128874


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {6.531128874, -1.531128874}
Solution
a = {6.531128874, -1.531128874}

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