0=-0.5x^2+7x-20

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Solution for 0=-0.5x^2+7x-20 equation: 


Simplifying
0 = -0.5x2 + 7x + -20

Reorder the terms:
0 = -20 + 7x + -0.5x2

Solving
0 = -20 + 7x + -0.5x2

Solving for variable 'x'.

Combine like terms: 0 + 20 = 20
20 + -7x + 0.5x2 = -20 + 7x + -0.5x2 + 20 + -7x + 0.5x2

Reorder the terms:
20 + -7x + 0.5x2 = -20 + 20 + 7x + -7x + -0.5x2 + 0.5x2

Combine like terms: -20 + 20 = 0
20 + -7x + 0.5x2 = 0 + 7x + -7x + -0.5x2 + 0.5x2
20 + -7x + 0.5x2 = 7x + -7x + -0.5x2 + 0.5x2

Combine like terms: 7x + -7x = 0
20 + -7x + 0.5x2 = 0 + -0.5x2 + 0.5x2
20 + -7x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
20 + -7x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
40 + -14x + x2 = 0

Move the constant term to the right:

Add '-40' to each side of the equation.
40 + -14x + -40 + x2 = 0 + -40

Reorder the terms:
40 + -40 + -14x + x2 = 0 + -40

Combine like terms: 40 + -40 = 0
0 + -14x + x2 = 0 + -40
-14x + x2 = 0 + -40

Combine like terms: 0 + -40 = -40
-14x + x2 = -40

The x term is -14x.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14x + 49 + x2 = -40 + 49

Reorder the terms:
49 + -14x + x2 = -40 + 49

Combine like terms: -40 + 49 = 9
49 + -14x + x2 = 9

Factor a perfect square on the left side:
(x + -7)(x + -7) = 9

Calculate the square root of the right side: 3

Break this problem into two subproblems by setting 
(x + -7) equal to 3 and -3.


Subproblem 1
x + -7 = 3

Simplifying
x + -7 = 3

Reorder the terms:
-7 + x = 3

Solving
-7 + x = 3

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '7' to each side of the equation.
-7 + 7 + x = 3 + 7

Combine like terms: -7 + 7 = 0
0 + x = 3 + 7
x = 3 + 7

Combine like terms: 3 + 7 = 10
x = 10

Simplifying
x = 10


Subproblem 2
x + -7 = -3

Simplifying
x + -7 = -3

Reorder the terms:
-7 + x = -3

Solving
-7 + x = -3

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '7' to each side of the equation.
-7 + 7 + x = -3 + 7

Combine like terms: -7 + 7 = 0
0 + x = -3 + 7
x = -3 + 7

Combine like terms: -3 + 7 = 4
x = 4

Simplifying
x = 4


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {10, 4}

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