x^2=21x-22

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Solution for x^2=21x-22 equation:


Simplifying
x2 = 21x + -22

Reorder the terms:
x2 = -22 + 21x

Solving
x2 = -22 + 21x

Solving for variable 'x'.

Reorder the terms:
22 + -21x + x2 = -22 + 21x + 22 + -21x

Reorder the terms:
22 + -21x + x2 = -22 + 22 + 21x + -21x

Combine like terms: -22 + 22 = 0
22 + -21x + x2 = 0 + 21x + -21x
22 + -21x + x2 = 21x + -21x

Combine like terms: 21x + -21x = 0
22 + -21x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -21x + -22 + x2 = 0 + -22

Reorder the terms:
22 + -22 + -21x + x2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -21x + x2 = 0 + -22
-21x + x2 = 0 + -22

Combine like terms: 0 + -22 = -22
-21x + x2 = -22

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = -22 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -22 + 110.25

Combine like terms: -22 + 110.25 = 88.25
110.25 + -21x + x2 = 88.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 88.25

Calculate the square root of the right side: 9.394147114

Break this problem into two subproblems by setting 
(x + -10.5) equal to 9.394147114 and -9.394147114.

Subproblem 1

x + -10.5 = 9.394147114 Simplifying x + -10.5 = 9.394147114 Reorder the terms: -10.5 + x = 9.394147114 Solving -10.5 + x = 9.394147114 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = 9.394147114 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 9.394147114 + 10.5 x = 9.394147114 + 10.5 Combine like terms: 9.394147114 + 10.5 = 19.894147114 x = 19.894147114 Simplifying x = 19.894147114

Subproblem 2

x + -10.5 = -9.394147114 Simplifying x + -10.5 = -9.394147114 Reorder the terms: -10.5 + x = -9.394147114 Solving -10.5 + x = -9.394147114 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = -9.394147114 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -9.394147114 + 10.5 x = -9.394147114 + 10.5 Combine like terms: -9.394147114 + 10.5 = 1.105852886 x = 1.105852886 Simplifying x = 1.105852886

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.894147114, 1.105852886}

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