2a[a+3]=3[a+2]

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Solution for 2a[a+3]=3[a+2] equation:


Simplifying
2a[a + 3] = 3[a + 2]

Reorder the terms:
2a[3 + a] = 3[a + 2]
[3 * 2a + a * 2a] = 3[a + 2]
[6a + 2a2] = 3[a + 2]

Reorder the terms:
6a + 2a2 = 3[2 + a]
6a + 2a2 = [2 * 3 + a * 3]
6a + 2a2 = [6 + 3a]

Solving
6a + 2a2 = 6 + 3a

Solving for variable 'a'.

Reorder the terms:
-6 + 6a + -3a + 2a2 = 6 + 3a + -6 + -3a

Combine like terms: 6a + -3a = 3a
-6 + 3a + 2a2 = 6 + 3a + -6 + -3a

Reorder the terms:
-6 + 3a + 2a2 = 6 + -6 + 3a + -3a

Combine like terms: 6 + -6 = 0
-6 + 3a + 2a2 = 0 + 3a + -3a
-6 + 3a + 2a2 = 3a + -3a

Combine like terms: 3a + -3a = 0
-6 + 3a + 2a2 = 0

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

Divide each side by '2'.
-3 + 1.5a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.5a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.5a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.5a + a2 = 0 + 3
1.5a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.5a + a2 = 3

The a term is 1.5a.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5a + 0.5625 + a2 = 3 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 3 + 0.5625

Combine like terms: 3 + 0.5625 = 3.5625
0.5625 + 1.5a + a2 = 3.5625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(a + 0.75) equal to 1.887458609 and -1.887458609.

Subproblem 1

a + 0.75 = 1.887458609 Simplifying a + 0.75 = 1.887458609 Reorder the terms: 0.75 + a = 1.887458609 Solving 0.75 + a = 1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 1.887458609 + -0.75 a = 1.887458609 + -0.75 Combine like terms: 1.887458609 + -0.75 = 1.137458609 a = 1.137458609 Simplifying a = 1.137458609

Subproblem 2

a + 0.75 = -1.887458609 Simplifying a + 0.75 = -1.887458609 Reorder the terms: 0.75 + a = -1.887458609 Solving 0.75 + a = -1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -1.887458609 + -0.75 a = -1.887458609 + -0.75 Combine like terms: -1.887458609 + -0.75 = -2.637458609 a = -2.637458609 Simplifying a = -2.637458609

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.137458609, -2.637458609}

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