2a(a-5)+4(a-5)=a

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Solution for 2a(a-5)+4(a-5)=a equation:


Simplifying
2a(a + -5) + 4(a + -5) = a

Reorder the terms:
2a(-5 + a) + 4(a + -5) = a
(-5 * 2a + a * 2a) + 4(a + -5) = a
(-10a + 2a2) + 4(a + -5) = a

Reorder the terms:
-10a + 2a2 + 4(-5 + a) = a
-10a + 2a2 + (-5 * 4 + a * 4) = a
-10a + 2a2 + (-20 + 4a) = a

Reorder the terms:
-20 + -10a + 4a + 2a2 = a

Combine like terms: -10a + 4a = -6a
-20 + -6a + 2a2 = a

Solving
-20 + -6a + 2a2 = a

Solving for variable 'a'.

Reorder the terms:
-20 + -6a + -1a + 2a2 = a + -1a

Combine like terms: -6a + -1a = -7a
-20 + -7a + 2a2 = a + -1a

Combine like terms: a + -1a = 0
-20 + -7a + 2a2 = 0

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

Divide each side by '2'.
-10 + -3.5a + a2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -3.5a + 10 + a2 = 0 + 10

Reorder the terms:
-10 + 10 + -3.5a + a2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -3.5a + a2 = 0 + 10
-3.5a + a2 = 0 + 10

Combine like terms: 0 + 10 = 10
-3.5a + a2 = 10

The a term is -3.5a.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5a + 3.0625 + a2 = 10 + 3.0625

Reorder the terms:
3.0625 + -3.5a + a2 = 10 + 3.0625

Combine like terms: 10 + 3.0625 = 13.0625
3.0625 + -3.5a + a2 = 13.0625

Factor a perfect square on the left side:
(a + -1.75)(a + -1.75) = 13.0625

Calculate the square root of the right side: 3.614208074

Break this problem into two subproblems by setting 
(a + -1.75) equal to 3.614208074 and -3.614208074.

Subproblem 1

a + -1.75 = 3.614208074 Simplifying a + -1.75 = 3.614208074 Reorder the terms: -1.75 + a = 3.614208074 Solving -1.75 + a = 3.614208074 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = 3.614208074 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = 3.614208074 + 1.75 a = 3.614208074 + 1.75 Combine like terms: 3.614208074 + 1.75 = 5.364208074 a = 5.364208074 Simplifying a = 5.364208074

Subproblem 2

a + -1.75 = -3.614208074 Simplifying a + -1.75 = -3.614208074 Reorder the terms: -1.75 + a = -3.614208074 Solving -1.75 + a = -3.614208074 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = -3.614208074 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = -3.614208074 + 1.75 a = -3.614208074 + 1.75 Combine like terms: -3.614208074 + 1.75 = -1.864208074 a = -1.864208074 Simplifying a = -1.864208074

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.364208074, -1.864208074}

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