x^2=10x-22

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Solution for x^2=10x-22 equation:


Simplifying
x2 = 10x + -22

Reorder the terms:
x2 = -22 + 10x

Solving
x2 = -22 + 10x

Solving for variable 'x'.

Reorder the terms:
22 + -10x + x2 = -22 + 10x + 22 + -10x

Reorder the terms:
22 + -10x + x2 = -22 + 22 + 10x + -10x

Combine like terms: -22 + 22 = 0
22 + -10x + x2 = 0 + 10x + -10x
22 + -10x + x2 = 10x + -10x

Combine like terms: 10x + -10x = 0
22 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -10x + -22 + x2 = 0 + -22

Reorder the terms:
22 + -22 + -10x + x2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -10x + x2 = 0 + -22
-10x + x2 = 0 + -22

Combine like terms: 0 + -22 = -22
-10x + x2 = -22

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -22 + 25

Reorder the terms:
25 + -10x + x2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + -10x + x2 = 3

Factor a perfect square on the left side:
(x + -5)(x + -5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -5) equal to 1.732050808 and -1.732050808.

Subproblem 1

x + -5 = 1.732050808 Simplifying x + -5 = 1.732050808 Reorder the terms: -5 + x = 1.732050808 Solving -5 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + x = 1.732050808 + 5 x = 1.732050808 + 5 Combine like terms: 1.732050808 + 5 = 6.732050808 x = 6.732050808 Simplifying x = 6.732050808

Subproblem 2

x + -5 = -1.732050808 Simplifying x + -5 = -1.732050808 Reorder the terms: -5 + x = -1.732050808 Solving -5 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + x = -1.732050808 + 5 x = -1.732050808 + 5 Combine like terms: -1.732050808 + 5 = 3.267949192 x = 3.267949192 Simplifying x = 3.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.732050808, 3.267949192}

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