2a(a-1)=3a+40

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Solution for 2a(a-1)=3a+40 equation:


Simplifying
2a(a + -1) = 3a + 40

Reorder the terms:
2a(-1 + a) = 3a + 40
(-1 * 2a + a * 2a) = 3a + 40
(-2a + 2a2) = 3a + 40

Reorder the terms:
-2a + 2a2 = 40 + 3a

Solving
-2a + 2a2 = 40 + 3a

Solving for variable 'a'.

Reorder the terms:
-40 + -2a + -3a + 2a2 = 40 + 3a + -40 + -3a

Combine like terms: -2a + -3a = -5a
-40 + -5a + 2a2 = 40 + 3a + -40 + -3a

Reorder the terms:
-40 + -5a + 2a2 = 40 + -40 + 3a + -3a

Combine like terms: 40 + -40 = 0
-40 + -5a + 2a2 = 0 + 3a + -3a
-40 + -5a + 2a2 = 3a + -3a

Combine like terms: 3a + -3a = 0
-40 + -5a + 2a2 = 0

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

Divide each side by '2'.
-20 + -2.5a + a2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -2.5a + 20 + a2 = 0 + 20

Reorder the terms:
-20 + 20 + -2.5a + a2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -2.5a + a2 = 0 + 20
-2.5a + a2 = 0 + 20

Combine like terms: 0 + 20 = 20
-2.5a + a2 = 20

The a term is -2.5a.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5a + 1.5625 + a2 = 20 + 1.5625

Reorder the terms:
1.5625 + -2.5a + a2 = 20 + 1.5625

Combine like terms: 20 + 1.5625 = 21.5625
1.5625 + -2.5a + a2 = 21.5625

Factor a perfect square on the left side:
(a + -1.25)(a + -1.25) = 21.5625

Calculate the square root of the right side: 4.643543905

Break this problem into two subproblems by setting 
(a + -1.25) equal to 4.643543905 and -4.643543905.

Subproblem 1

a + -1.25 = 4.643543905 Simplifying a + -1.25 = 4.643543905 Reorder the terms: -1.25 + a = 4.643543905 Solving -1.25 + a = 4.643543905 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 4.643543905 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 4.643543905 + 1.25 a = 4.643543905 + 1.25 Combine like terms: 4.643543905 + 1.25 = 5.893543905 a = 5.893543905 Simplifying a = 5.893543905

Subproblem 2

a + -1.25 = -4.643543905 Simplifying a + -1.25 = -4.643543905 Reorder the terms: -1.25 + a = -4.643543905 Solving -1.25 + a = -4.643543905 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -4.643543905 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -4.643543905 + 1.25 a = -4.643543905 + 1.25 Combine like terms: -4.643543905 + 1.25 = -3.393543905 a = -3.393543905 Simplifying a = -3.393543905

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.893543905, -3.393543905}

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