0=x^2-2.5x+1

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Solution for 0=x^2-2.5x+1 equation:


Simplifying
0 = x2 + -2.5x + 1

Reorder the terms:
0 = 1 + -2.5x + x2

Solving
0 = 1 + -2.5x + x2

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 2.5x + -1x2 = 1 + -2.5x + x2 + -1 + 2.5x + -1x2

Reorder the terms:
-1 + 2.5x + -1x2 = 1 + -1 + -2.5x + 2.5x + x2 + -1x2

Combine like terms: 1 + -1 = 0
-1 + 2.5x + -1x2 = 0 + -2.5x + 2.5x + x2 + -1x2
-1 + 2.5x + -1x2 = -2.5x + 2.5x + x2 + -1x2

Combine like terms: -2.5x + 2.5x = 0.0
-1 + 2.5x + -1x2 = 0.0 + x2 + -1x2
-1 + 2.5x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + 2.5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.5x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -2.5x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.5x + x2 = 0 + -1
-2.5x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.5x + x2 = -1

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = -1 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -1 + 1.5625

Combine like terms: -1 + 1.5625 = 0.5625
1.5625 + -2.5x + x2 = 0.5625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 0.5625

Calculate the square root of the right side: 0.75

Break this problem into two subproblems by setting 
(x + -1.25) equal to 0.75 and -0.75.

Subproblem 1

x + -1.25 = 0.75 Simplifying x + -1.25 = 0.75 Reorder the terms: -1.25 + x = 0.75 Solving -1.25 + x = 0.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 0.75 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.75 + 1.25 x = 0.75 + 1.25 Combine like terms: 0.75 + 1.25 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + -1.25 = -0.75 Simplifying x + -1.25 = -0.75 Reorder the terms: -1.25 + x = -0.75 Solving -1.25 + x = -0.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -0.75 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.75 + 1.25 x = -0.75 + 1.25 Combine like terms: -0.75 + 1.25 = 0.5 x = 0.5 Simplifying x = 0.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, 0.5}

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