105=0.5x*x+0.5x

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Solution for 105=0.5x*x+0.5x equation:


Simplifying
105 = 0.5x * x + 0.5x

Multiply x * x
105 = 0.5x2 + 0.5x

Reorder the terms:
105 = 0.5x + 0.5x2

Solving
105 = 0.5x + 0.5x2

Solving for variable 'x'.

Reorder the terms:
105 + -0.5x + -0.5x2 = 0.5x + -0.5x + 0.5x2 + -0.5x2

Combine like terms: 0.5x + -0.5x = 0.0
105 + -0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
105 + -0.5x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
105 + -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'.
-210 + x + x2 = 0

Move the constant term to the right:

Add '210' to each side of the equation.
-210 + x + 210 + x2 = 0 + 210

Reorder the terms:
-210 + 210 + x + x2 = 0 + 210

Combine like terms: -210 + 210 = 0
0 + x + x2 = 0 + 210
x + x2 = 0 + 210

Combine like terms: 0 + 210 = 210
x + x2 = 210

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 = 210 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 210 + 0.25

Combine like terms: 210 + 0.25 = 210.25
1.25 + x2 = 210.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 210.25

Calculate the square root of the right side: 14.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 14.5 and -14.5.

Subproblem 1

x + 0.5 = 14.5 Simplifying x + 0.5 = 14.5 Reorder the terms: 0.5 + x = 14.5 Solving 0.5 + x = 14.5 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 = 14.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 14.5 + -0.5 x = 14.5 + -0.5 Combine like terms: 14.5 + -0.5 = 14 x = 14 Simplifying x = 14

Subproblem 2

x + 0.5 = -14.5 Simplifying x + 0.5 = -14.5 Reorder the terms: 0.5 + x = -14.5 Solving 0.5 + x = -14.5 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 = -14.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -14.5 + -0.5 x = -14.5 + -0.5 Combine like terms: -14.5 + -0.5 = -15 x = -15 Simplifying x = -15

Solution

The solution to the problem is based on the solutions from the subproblems. x = {14, -15}

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