6.75=0.5x^2-0.5x+0.75

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Solution for 6.75=0.5x^2-0.5x+0.75 equation:


Simplifying
6.75 = 0.5x2 + -0.5x + 0.75

Reorder the terms:
6.75 = 0.75 + -0.5x + 0.5x2

Solving
6.75 = 0.75 + -0.5x + 0.5x2

Solving for variable 'x'.

Combine like terms: 6.75 + -0.75 = 6
6 + 0.5x + -0.5x2 = 0.75 + -0.5x + 0.5x2 + -0.75 + 0.5x + -0.5x2

Reorder the terms:
6 + 0.5x + -0.5x2 = 0.75 + -0.75 + -0.5x + 0.5x + 0.5x2 + -0.5x2

Combine like terms: 0.75 + -0.75 = 0.00
6 + 0.5x + -0.5x2 = 0.00 + -0.5x + 0.5x + 0.5x2 + -0.5x2
6 + 0.5x + -0.5x2 = -0.5x + 0.5x + 0.5x2 + -0.5x2

Combine like terms: -0.5x + 0.5x = 0.0
6 + 0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
6 + 0.5x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
6 + 0.5x + -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'.
-12 + -1x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -1x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + -1x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -1x + x2 = 0 + 12
-1x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
-1x + x2 = 12

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 12 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 12 + 0.25

Combine like terms: 12 + 0.25 = 12.25
0.25 + -1x + x2 = 12.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(x + -0.5) equal to 3.5 and -3.5.

Subproblem 1

x + -0.5 = 3.5 Simplifying x + -0.5 = 3.5 Reorder the terms: -0.5 + x = 3.5 Solving -0.5 + x = 3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 3.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.5 + 0.5 x = 3.5 + 0.5 Combine like terms: 3.5 + 0.5 = 4 x = 4 Simplifying x = 4

Subproblem 2

x + -0.5 = -3.5 Simplifying x + -0.5 = -3.5 Reorder the terms: -0.5 + x = -3.5 Solving -0.5 + x = -3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -3.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.5 + 0.5 x = -3.5 + 0.5 Combine like terms: -3.5 + 0.5 = -3 x = -3 Simplifying x = -3

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4, -3}

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