0.5x+0.5x^2=6

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Solution for 0.5x+0.5x^2=6 equation:


Simplifying
0.5x + 0.5x2 = 6

Solving
0.5x + 0.5x2 = 6

Solving for variable 'x'.

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

Combine like terms: 6 + -6 = 0
-6 + 0.5x + 0.5x2 = 0

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

Divide each side by '0.5'.
-12 + x + x2 = 0

Move the constant term to the right:

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

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

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

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

The x term is x.  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.
 + 0.25 + x2 = 12 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 12 + 0.25

Combine like terms: 12 + 0.25 = 12.25
1.25 + 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 = 3 x = 3 Simplifying x = 3

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 = -4 x = -4 Simplifying x = -4

Solution

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

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