0=x^2-32x+14

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Solution for 0=x^2-32x+14 equation:


Simplifying
0 = x2 + -32x + 14

Reorder the terms:
0 = 14 + -32x + x2

Solving
0 = 14 + -32x + x2

Solving for variable 'x'.

Combine like terms: 0 + -14 = -14
-14 + 32x + -1x2 = 14 + -32x + x2 + -14 + 32x + -1x2

Reorder the terms:
-14 + 32x + -1x2 = 14 + -14 + -32x + 32x + x2 + -1x2

Combine like terms: 14 + -14 = 0
-14 + 32x + -1x2 = 0 + -32x + 32x + x2 + -1x2
-14 + 32x + -1x2 = -32x + 32x + x2 + -1x2

Combine like terms: -32x + 32x = 0
-14 + 32x + -1x2 = 0 + x2 + -1x2
-14 + 32x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-14 + 32x + -1x2 = 0

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

Divide each side by '-1'.
14 + -32x + x2 = 0

Move the constant term to the right:

Add '-14' to each side of the equation.
14 + -32x + -14 + x2 = 0 + -14

Reorder the terms:
14 + -14 + -32x + x2 = 0 + -14

Combine like terms: 14 + -14 = 0
0 + -32x + x2 = 0 + -14
-32x + x2 = 0 + -14

Combine like terms: 0 + -14 = -14
-32x + x2 = -14

The x term is -32x.  Take half its coefficient (-16).
Square it (256) and add it to both sides.

Add '256' to each side of the equation.
-32x + 256 + x2 = -14 + 256

Reorder the terms:
256 + -32x + x2 = -14 + 256

Combine like terms: -14 + 256 = 242
256 + -32x + x2 = 242

Factor a perfect square on the left side:
(x + -16)(x + -16) = 242

Calculate the square root of the right side: 15.556349186

Break this problem into two subproblems by setting 
(x + -16) equal to 15.556349186 and -15.556349186.

Subproblem 1

x + -16 = 15.556349186 Simplifying x + -16 = 15.556349186 Reorder the terms: -16 + x = 15.556349186 Solving -16 + x = 15.556349186 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x = 15.556349186 + 16 Combine like terms: -16 + 16 = 0 0 + x = 15.556349186 + 16 x = 15.556349186 + 16 Combine like terms: 15.556349186 + 16 = 31.556349186 x = 31.556349186 Simplifying x = 31.556349186

Subproblem 2

x + -16 = -15.556349186 Simplifying x + -16 = -15.556349186 Reorder the terms: -16 + x = -15.556349186 Solving -16 + x = -15.556349186 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x = -15.556349186 + 16 Combine like terms: -16 + 16 = 0 0 + x = -15.556349186 + 16 x = -15.556349186 + 16 Combine like terms: -15.556349186 + 16 = 0.443650814 x = 0.443650814 Simplifying x = 0.443650814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {31.556349186, 0.443650814}

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