-x^2+21x-28=0

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Solution for -x^2+21x-28=0 equation: 


Simplifying
-1x2 + 21x + -28 = 0

Reorder the terms:
-28 + 21x + -1x2 = 0

Solving
-28 + 21x + -1x2 = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
28 + -21x + x2 = 0

Move the constant term to the right:

Add '-28' to each side of the equation.
28 + -21x + -28 + x2 = 0 + -28

Reorder the terms:
28 + -28 + -21x + x2 = 0 + -28

Combine like terms: 28 + -28 = 0
0 + -21x + x2 = 0 + -28
-21x + x2 = 0 + -28

Combine like terms: 0 + -28 = -28
-21x + x2 = -28

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 = -28 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -28 + 110.25

Combine like terms: -28 + 110.25 = 82.25
110.25 + -21x + x2 = 82.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(x + -10.5) equal to 9.069178574 and -9.069178574.


Subproblem 1
x + -10.5 = 9.069178574

Simplifying
x + -10.5 = 9.069178574

Reorder the terms:
-10.5 + x = 9.069178574

Solving
-10.5 + x = 9.069178574

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 = 9.069178574 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = 9.069178574 + 10.5
x = 9.069178574 + 10.5

Combine like terms: 9.069178574 + 10.5 = 19.569178574
x = 19.569178574

Simplifying
x = 19.569178574


Subproblem 2
x + -10.5 = -9.069178574

Simplifying
x + -10.5 = -9.069178574

Reorder the terms:
-10.5 + x = -9.069178574

Solving
-10.5 + x = -9.069178574

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 = -9.069178574 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = -9.069178574 + 10.5
x = -9.069178574 + 10.5

Combine like terms: -9.069178574 + 10.5 = 1.430821426
x = 1.430821426

Simplifying
x = 1.430821426


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.569178574, 1.430821426}

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