0=x^2-31x+74

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Solution for 0=x^2-31x+74 equation:


Simplifying
0 = x2 + -31x + 74

Reorder the terms:
0 = 74 + -31x + x2

Solving
0 = 74 + -31x + x2

Solving for variable 'x'.

Combine like terms: 0 + -74 = -74
-74 + 31x + -1x2 = 74 + -31x + x2 + -74 + 31x + -1x2

Reorder the terms:
-74 + 31x + -1x2 = 74 + -74 + -31x + 31x + x2 + -1x2

Combine like terms: 74 + -74 = 0
-74 + 31x + -1x2 = 0 + -31x + 31x + x2 + -1x2
-74 + 31x + -1x2 = -31x + 31x + x2 + -1x2

Combine like terms: -31x + 31x = 0
-74 + 31x + -1x2 = 0 + x2 + -1x2
-74 + 31x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-74 + 31x + -1x2 = 0

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

Divide each side by '-1'.
74 + -31x + x2 = 0

Move the constant term to the right:

Add '-74' to each side of the equation.
74 + -31x + -74 + x2 = 0 + -74

Reorder the terms:
74 + -74 + -31x + x2 = 0 + -74

Combine like terms: 74 + -74 = 0
0 + -31x + x2 = 0 + -74
-31x + x2 = 0 + -74

Combine like terms: 0 + -74 = -74
-31x + x2 = -74

The x term is -31x.  Take half its coefficient (-15.5).
Square it (240.25) and add it to both sides.

Add '240.25' to each side of the equation.
-31x + 240.25 + x2 = -74 + 240.25

Reorder the terms:
240.25 + -31x + x2 = -74 + 240.25

Combine like terms: -74 + 240.25 = 166.25
240.25 + -31x + x2 = 166.25

Factor a perfect square on the left side:
(x + -15.5)(x + -15.5) = 166.25

Calculate the square root of the right side: 12.893796958

Break this problem into two subproblems by setting 
(x + -15.5) equal to 12.893796958 and -12.893796958.

Subproblem 1

x + -15.5 = 12.893796958 Simplifying x + -15.5 = 12.893796958 Reorder the terms: -15.5 + x = 12.893796958 Solving -15.5 + x = 12.893796958 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15.5' to each side of the equation. -15.5 + 15.5 + x = 12.893796958 + 15.5 Combine like terms: -15.5 + 15.5 = 0.0 0.0 + x = 12.893796958 + 15.5 x = 12.893796958 + 15.5 Combine like terms: 12.893796958 + 15.5 = 28.393796958 x = 28.393796958 Simplifying x = 28.393796958

Subproblem 2

x + -15.5 = -12.893796958 Simplifying x + -15.5 = -12.893796958 Reorder the terms: -15.5 + x = -12.893796958 Solving -15.5 + x = -12.893796958 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15.5' to each side of the equation. -15.5 + 15.5 + x = -12.893796958 + 15.5 Combine like terms: -15.5 + 15.5 = 0.0 0.0 + x = -12.893796958 + 15.5 x = -12.893796958 + 15.5 Combine like terms: -12.893796958 + 15.5 = 2.606203042 x = 2.606203042 Simplifying x = 2.606203042

Solution

The solution to the problem is based on the solutions from the subproblems. x = {28.393796958, 2.606203042}

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