0.25x^2+1=-0.5x^2+2x+0.5

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Solution for 0.25x^2+1=-0.5x^2+2x+0.5 equation:


Simplifying
0.25x2 + 1 = -0.5x2 + 2x + 0.5

Reorder the terms:
1 + 0.25x2 = -0.5x2 + 2x + 0.5

Reorder the terms:
1 + 0.25x2 = 0.5 + 2x + -0.5x2

Solving
1 + 0.25x2 = 0.5 + 2x + -0.5x2

Solving for variable 'x'.

Reorder the terms:
1 + -0.5 + -2x + 0.25x2 + 0.5x2 = 0.5 + 2x + -0.5x2 + -0.5 + -2x + 0.5x2

Combine like terms: 1 + -0.5 = 0.5
0.5 + -2x + 0.25x2 + 0.5x2 = 0.5 + 2x + -0.5x2 + -0.5 + -2x + 0.5x2

Combine like terms: 0.25x2 + 0.5x2 = 0.75x2
0.5 + -2x + 0.75x2 = 0.5 + 2x + -0.5x2 + -0.5 + -2x + 0.5x2

Reorder the terms:
0.5 + -2x + 0.75x2 = 0.5 + -0.5 + 2x + -2x + -0.5x2 + 0.5x2

Combine like terms: 0.5 + -0.5 = 0.0
0.5 + -2x + 0.75x2 = 0.0 + 2x + -2x + -0.5x2 + 0.5x2
0.5 + -2x + 0.75x2 = 2x + -2x + -0.5x2 + 0.5x2

Combine like terms: 2x + -2x = 0
0.5 + -2x + 0.75x2 = 0 + -0.5x2 + 0.5x2
0.5 + -2x + 0.75x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
0.5 + -2x + 0.75x2 = 0.0

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

Divide each side by '0.75'.
0.6666666667 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -2.666666667x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -2.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + -0.6666666667
-2.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-2.666666667x + x2 = -0.6666666667

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = -0.6666666667 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -0.6666666667 + 1.777777780

Combine like terms: -0.6666666667 + 1.777777780 = 1.1111111133
1.777777780 + -2.666666667x + x2 = 1.1111111133

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 1.1111111133

Calculate the square root of the right side: 1.054092554

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.054092554 and -1.054092554.

Subproblem 1

x + -1.333333334 = 1.054092554 Simplifying x + -1.333333334 = 1.054092554 Reorder the terms: -1.333333334 + x = 1.054092554 Solving -1.333333334 + x = 1.054092554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.054092554 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.054092554 + 1.333333334 x = 1.054092554 + 1.333333334 Combine like terms: 1.054092554 + 1.333333334 = 2.387425888 x = 2.387425888 Simplifying x = 2.387425888

Subproblem 2

x + -1.333333334 = -1.054092554 Simplifying x + -1.333333334 = -1.054092554 Reorder the terms: -1.333333334 + x = -1.054092554 Solving -1.333333334 + x = -1.054092554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.054092554 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.054092554 + 1.333333334 x = -1.054092554 + 1.333333334 Combine like terms: -1.054092554 + 1.333333334 = 0.27924078 x = 0.27924078 Simplifying x = 0.27924078

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.387425888, 0.27924078}

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