.5x-.5x^2=-1

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Solution for .5x-.5x^2=-1 equation:


Simplifying
0.5x + -0.5x2 = -1

Solving
0.5x + -0.5x2 = -1

Solving for variable 'x'.

Reorder the terms:
1 + 0.5x + -0.5x2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + 0.5x + -0.5x2 = 0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-2 + -1x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -1x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1x + x2 = 0 + 2
-1x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1x + x2 = 2

The x term is -1x.  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.
-1x + 0.25 + x2 = 2 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + -1x + x2 = 2.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.5 and -1.5.

Subproblem 1

x + -0.5 = 1.5 Simplifying x + -0.5 = 1.5 Reorder the terms: -0.5 + x = 1.5 Solving -0.5 + x = 1.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.5 + 0.5 x = 1.5 + 0.5 Combine like terms: 1.5 + 0.5 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + -0.5 = -1.5 Simplifying x + -0.5 = -1.5 Reorder the terms: -0.5 + x = -1.5 Solving -0.5 + x = -1.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.5 + 0.5 x = -1.5 + 0.5 Combine like terms: -1.5 + 0.5 = -1 x = -1 Simplifying x = -1

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, -1}

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