x^2-1=0.5x+0.5

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


Simplifying
x2 + -1 = 0.5x + 0.5

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

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

Solving
-1 + x2 = 0.5 + 0.5x

Solving for variable 'x'.

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

Combine like terms: -1 + -0.5 = -1.5
-1.5 + -0.5x + x2 = 0.5 + 0.5x + -0.5 + -0.5x

Reorder the terms:
-1.5 + -0.5x + x2 = 0.5 + -0.5 + 0.5x + -0.5x

Combine like terms: 0.5 + -0.5 = 0.0
-1.5 + -0.5x + x2 = 0.0 + 0.5x + -0.5x
-1.5 + -0.5x + x2 = 0.5x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
-1.5 + -0.5x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.5x + 1.5 + x2 = 0.0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.5x + x2 = 0.0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.5x + x2 = 0.0 + 1.5
-0.5x + x2 = 0.0 + 1.5

Combine like terms: 0.0 + 1.5 = 1.5
-0.5x + x2 = 1.5

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 = 1.5 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.5 + 0.0625

Combine like terms: 1.5 + 0.0625 = 1.5625
0.0625 + -0.5x + x2 = 1.5625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.5625

Calculate the square root of the right side: 1.25

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.25 and -1.25.

Subproblem 1

x + -0.25 = 1.25 Simplifying x + -0.25 = 1.25 Reorder the terms: -0.25 + x = 1.25 Solving -0.25 + x = 1.25 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 = 1.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.25 + 0.25 x = 1.25 + 0.25 Combine like terms: 1.25 + 0.25 = 1.5 x = 1.5 Simplifying x = 1.5

Subproblem 2

x + -0.25 = -1.25 Simplifying x + -0.25 = -1.25 Reorder the terms: -0.25 + x = -1.25 Solving -0.25 + x = -1.25 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 = -1.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.25 + 0.25 x = -1.25 + 0.25 Combine like terms: -1.25 + 0.25 = -1 x = -1 Simplifying x = -1

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5, -1}

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