0=x^2-25x+120

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Solution for 0=x^2-25x+120 equation:


Simplifying
0 = x2 + -25x + 120

Reorder the terms:
0 = 120 + -25x + x2

Solving
0 = 120 + -25x + x2

Solving for variable 'x'.

Combine like terms: 0 + -120 = -120
-120 + 25x + -1x2 = 120 + -25x + x2 + -120 + 25x + -1x2

Reorder the terms:
-120 + 25x + -1x2 = 120 + -120 + -25x + 25x + x2 + -1x2

Combine like terms: 120 + -120 = 0
-120 + 25x + -1x2 = 0 + -25x + 25x + x2 + -1x2
-120 + 25x + -1x2 = -25x + 25x + x2 + -1x2

Combine like terms: -25x + 25x = 0
-120 + 25x + -1x2 = 0 + x2 + -1x2
-120 + 25x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-120 + 25x + -1x2 = 0

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

Divide each side by '-1'.
120 + -25x + x2 = 0

Move the constant term to the right:

Add '-120' to each side of the equation.
120 + -25x + -120 + x2 = 0 + -120

Reorder the terms:
120 + -120 + -25x + x2 = 0 + -120

Combine like terms: 120 + -120 = 0
0 + -25x + x2 = 0 + -120
-25x + x2 = 0 + -120

Combine like terms: 0 + -120 = -120
-25x + x2 = -120

The x term is -25x.  Take half its coefficient (-12.5).
Square it (156.25) and add it to both sides.

Add '156.25' to each side of the equation.
-25x + 156.25 + x2 = -120 + 156.25

Reorder the terms:
156.25 + -25x + x2 = -120 + 156.25

Combine like terms: -120 + 156.25 = 36.25
156.25 + -25x + x2 = 36.25

Factor a perfect square on the left side:
(x + -12.5)(x + -12.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(x + -12.5) equal to 6.020797289 and -6.020797289.

Subproblem 1

x + -12.5 = 6.020797289 Simplifying x + -12.5 = 6.020797289 Reorder the terms: -12.5 + x = 6.020797289 Solving -12.5 + x = 6.020797289 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x = 6.020797289 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x = 6.020797289 + 12.5 x = 6.020797289 + 12.5 Combine like terms: 6.020797289 + 12.5 = 18.520797289 x = 18.520797289 Simplifying x = 18.520797289

Subproblem 2

x + -12.5 = -6.020797289 Simplifying x + -12.5 = -6.020797289 Reorder the terms: -12.5 + x = -6.020797289 Solving -12.5 + x = -6.020797289 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x = -6.020797289 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x = -6.020797289 + 12.5 x = -6.020797289 + 12.5 Combine like terms: -6.020797289 + 12.5 = 6.479202711 x = 6.479202711 Simplifying x = 6.479202711

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.520797289, 6.479202711}

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