0.9=2a-a^2

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Solution for 0.9=2a-a^2 equation:


Simplifying
0.9 = 2a + -1a2

Solving
0.9 = 2a + -1a2

Solving for variable 'a'.

Reorder the terms:
0.9 + -2a + a2 = 2a + -2a + -1a2 + a2

Combine like terms: 2a + -2a = 0
0.9 + -2a + a2 = 0 + -1a2 + a2
0.9 + -2a + a2 = -1a2 + a2

Combine like terms: -1a2 + a2 = 0
0.9 + -2a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-0.9' to each side of the equation.
0.9 + -2a + -0.9 + a2 = 0 + -0.9

Reorder the terms:
0.9 + -0.9 + -2a + a2 = 0 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + -2a + a2 = 0 + -0.9
-2a + a2 = 0 + -0.9

Combine like terms: 0 + -0.9 = -0.9
-2a + a2 = -0.9

The a term is -2a.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2a + 1 + a2 = -0.9 + 1

Reorder the terms:
1 + -2a + a2 = -0.9 + 1

Combine like terms: -0.9 + 1 = 0.1
1 + -2a + a2 = 0.1

Factor a perfect square on the left side:
(a + -1)(a + -1) = 0.1

Calculate the square root of the right side: 0.316227766

Break this problem into two subproblems by setting 
(a + -1) equal to 0.316227766 and -0.316227766.

Subproblem 1

a + -1 = 0.316227766 Simplifying a + -1 = 0.316227766 Reorder the terms: -1 + a = 0.316227766 Solving -1 + a = 0.316227766 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = 0.316227766 + 1 Combine like terms: -1 + 1 = 0 0 + a = 0.316227766 + 1 a = 0.316227766 + 1 Combine like terms: 0.316227766 + 1 = 1.316227766 a = 1.316227766 Simplifying a = 1.316227766

Subproblem 2

a + -1 = -0.316227766 Simplifying a + -1 = -0.316227766 Reorder the terms: -1 + a = -0.316227766 Solving -1 + a = -0.316227766 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = -0.316227766 + 1 Combine like terms: -1 + 1 = 0 0 + a = -0.316227766 + 1 a = -0.316227766 + 1 Combine like terms: -0.316227766 + 1 = 0.683772234 a = 0.683772234 Simplifying a = 0.683772234

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.316227766, 0.683772234}

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