x+.5(x-1)x=105

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Solution for x+.5(x-1)x=105 equation:


Simplifying
x + 0.5(x + -1) * x = 105

Reorder the terms:
x + 0.5(-1 + x) * x = 105

Reorder the terms for easier multiplication:
x + 0.5x(-1 + x) = 105
x + (-1 * 0.5x + x * 0.5x) = 105
x + (-0.5x + 0.5x2) = 105

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

Solving
0.5x + 0.5x2 = 105

Solving for variable 'x'.

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

Combine like terms: 105 + -105 = 0
-105 + 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'.
-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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