120-6x=0.5x^2+x

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Solution for 120-6x=0.5x^2+x equation:


Simplifying
120 + -6x = 0.5x2 + x

Reorder the terms:
120 + -6x = x + 0.5x2

Solving
120 + -6x = x + 0.5x2

Solving for variable 'x'.

Combine like terms: -6x + -1x = -7x
120 + -7x + -0.5x2 = x + 0.5x2 + -1x + -0.5x2

Reorder the terms:
120 + -7x + -0.5x2 = x + -1x + 0.5x2 + -0.5x2

Combine like terms: x + -1x = 0
120 + -7x + -0.5x2 = 0 + 0.5x2 + -0.5x2
120 + -7x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
120 + -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'.
-240 + 14x + x2 = 0

Move the constant term to the right:

Add '240' to each side of the equation.
-240 + 14x + 240 + x2 = 0 + 240

Reorder the terms:
-240 + 240 + 14x + x2 = 0 + 240

Combine like terms: -240 + 240 = 0
0 + 14x + x2 = 0 + 240
14x + x2 = 0 + 240

Combine like terms: 0 + 240 = 240
14x + x2 = 240

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 = 240 + 49

Reorder the terms:
49 + 14x + x2 = 240 + 49

Combine like terms: 240 + 49 = 289
49 + 14x + x2 = 289

Factor a perfect square on the left side:
(x + 7)(x + 7) = 289

Calculate the square root of the right side: 17

Break this problem into two subproblems by setting 
(x + 7) equal to 17 and -17.

Subproblem 1

x + 7 = 17 Simplifying x + 7 = 17 Reorder the terms: 7 + x = 17 Solving 7 + x = 17 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 = 17 + -7 Combine like terms: 7 + -7 = 0 0 + x = 17 + -7 x = 17 + -7 Combine like terms: 17 + -7 = 10 x = 10 Simplifying x = 10

Subproblem 2

x + 7 = -17 Simplifying x + 7 = -17 Reorder the terms: 7 + x = -17 Solving 7 + x = -17 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 = -17 + -7 Combine like terms: 7 + -7 = 0 0 + x = -17 + -7 x = -17 + -7 Combine like terms: -17 + -7 = -24 x = -24 Simplifying x = -24

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10, -24}

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