0=x^2-8x+13

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Solution for 0=x^2-8x+13 equation:


Simplifying
0 = x2 + -8x + 13

Reorder the terms:
0 = 13 + -8x + x2

Solving
0 = 13 + -8x + x2

Solving for variable 'x'.

Combine like terms: 0 + -13 = -13
-13 + 8x + -1x2 = 13 + -8x + x2 + -13 + 8x + -1x2

Reorder the terms:
-13 + 8x + -1x2 = 13 + -13 + -8x + 8x + x2 + -1x2

Combine like terms: 13 + -13 = 0
-13 + 8x + -1x2 = 0 + -8x + 8x + x2 + -1x2
-13 + 8x + -1x2 = -8x + 8x + x2 + -1x2

Combine like terms: -8x + 8x = 0
-13 + 8x + -1x2 = 0 + x2 + -1x2
-13 + 8x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-13 + 8x + -1x2 = 0

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

Divide each side by '-1'.
13 + -8x + x2 = 0

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -8x + -13 + x2 = 0 + -13

Reorder the terms:
13 + -13 + -8x + x2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -8x + x2 = 0 + -13
-8x + x2 = 0 + -13

Combine like terms: 0 + -13 = -13
-8x + x2 = -13

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = -13 + 16

Reorder the terms:
16 + -8x + x2 = -13 + 16

Combine like terms: -13 + 16 = 3
16 + -8x + x2 = 3

Factor a perfect square on the left side:
(x + -4)(x + -4) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -4) equal to 1.732050808 and -1.732050808.

Subproblem 1

x + -4 = 1.732050808 Simplifying x + -4 = 1.732050808 Reorder the terms: -4 + x = 1.732050808 Solving -4 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x = 1.732050808 + 4 x = 1.732050808 + 4 Combine like terms: 1.732050808 + 4 = 5.732050808 x = 5.732050808 Simplifying x = 5.732050808

Subproblem 2

x + -4 = -1.732050808 Simplifying x + -4 = -1.732050808 Reorder the terms: -4 + x = -1.732050808 Solving -4 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + x = -1.732050808 + 4 x = -1.732050808 + 4 Combine like terms: -1.732050808 + 4 = 2.267949192 x = 2.267949192 Simplifying x = 2.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.732050808, 2.267949192}

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