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

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


Simplifying
a(a + -2) + -1(3 + -1a2) = 2

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

Reorder the terms:
-3 + -2a + a2 + 1a2 = 2

Combine like terms: a2 + 1a2 = 2a2
-3 + -2a + 2a2 = 2

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

Solving for variable 'a'.

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

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

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

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

Divide each side by '2'.
-2.5 + -1a + a2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -1a + 2.5 + a2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -1a + a2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -1a + a2 = 0 + 2.5
-1a + a2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-1a + a2 = 2.5

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 = 2.5 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 2.5 + 0.25

Combine like terms: 2.5 + 0.25 = 2.75
0.25 + -1a + a2 = 2.75

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(a + -0.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

a + -0.5 = 1.658312395 Simplifying a + -0.5 = 1.658312395 Reorder the terms: -0.5 + a = 1.658312395 Solving -0.5 + a = 1.658312395 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 = 1.658312395 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 1.658312395 + 0.5 a = 1.658312395 + 0.5 Combine like terms: 1.658312395 + 0.5 = 2.158312395 a = 2.158312395 Simplifying a = 2.158312395

Subproblem 2

a + -0.5 = -1.658312395 Simplifying a + -0.5 = -1.658312395 Reorder the terms: -0.5 + a = -1.658312395 Solving -0.5 + a = -1.658312395 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 = -1.658312395 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -1.658312395 + 0.5 a = -1.658312395 + 0.5 Combine like terms: -1.658312395 + 0.5 = -1.158312395 a = -1.158312395 Simplifying a = -1.158312395

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.158312395, -1.158312395}

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