x^2+0.5x-1=0

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Solution for x^2+0.5x-1=0 equation:


Simplifying
x2 + 0.5x + -1 = 0

Reorder the terms:
-1 + 0.5x + x2 = 0

Solving
-1 + 0.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.5x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.5x + x2 = 0 + 1

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

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

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 1 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + 0.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

x + 0.25 = 1.030776406 Simplifying x + 0.25 = 1.030776406 Reorder the terms: 0.25 + x = 1.030776406 Solving 0.25 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.030776406 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.030776406 + -0.25 x = 1.030776406 + -0.25 Combine like terms: 1.030776406 + -0.25 = 0.780776406 x = 0.780776406 Simplifying x = 0.780776406

Subproblem 2

x + 0.25 = -1.030776406 Simplifying x + 0.25 = -1.030776406 Reorder the terms: 0.25 + x = -1.030776406 Solving 0.25 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.030776406 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.030776406 + -0.25 x = -1.030776406 + -0.25 Combine like terms: -1.030776406 + -0.25 = -1.280776406 x = -1.280776406 Simplifying x = -1.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.780776406, -1.280776406}

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