.25x^2+x=1

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Solution for .25x^2+x=1 equation:


Simplifying
0.25x2 + x = 1

Reorder the terms:
x + 0.25x2 = 1

Solving
x + 0.25x2 = 1

Solving for variable 'x'.

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

Combine like terms: 1 + -1 = 0
-1 + x + 0.25x2 = 0

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

Divide each side by '0.25'.
-4 + 4x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 4x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 4x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 4x + x2 = 0 + 4
4x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
4x + x2 = 4

The x term is x.  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.
4x + 0.25 + x2 = 4 + 0.25

Reorder the terms:
0.25 + 4x + x2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + 4x + x2 = 4.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x + 0.5 = 2.061552813 Simplifying x + 0.5 = 2.061552813 Reorder the terms: 0.5 + x = 2.061552813 Solving 0.5 + x = 2.061552813 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 = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.061552813 + -0.5 x = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 x = 1.561552813 Simplifying x = 1.561552813

Subproblem 2

x + 0.5 = -2.061552813 Simplifying x + 0.5 = -2.061552813 Reorder the terms: 0.5 + x = -2.061552813 Solving 0.5 + x = -2.061552813 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 = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.061552813 + -0.5 x = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 x = -2.561552813 Simplifying x = -2.561552813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.561552813, -2.561552813}

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