4x+0.5x^2=120

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Solution for 4x+0.5x^2=120 equation:


Simplifying
4x + 0.5x2 = 120

Solving
4x + 0.5x2 = 120

Solving for variable 'x'.

Reorder the terms:
-120 + 4x + 0.5x2 = 120 + -120

Combine like terms: 120 + -120 = 0
-120 + 4x + 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'.
-240 + 8x + x2 = 0

Move the constant term to the right:

Add '240' to each side of the equation.
-240 + 8x + 240 + x2 = 0 + 240

Reorder the terms:
-240 + 240 + 8x + x2 = 0 + 240

Combine like terms: -240 + 240 = 0
0 + 8x + x2 = 0 + 240
8x + x2 = 0 + 240

Combine like terms: 0 + 240 = 240
8x + x2 = 240

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = 240 + 16

Reorder the terms:
16 + 8x + x2 = 240 + 16

Combine like terms: 240 + 16 = 256
16 + 8x + x2 = 256

Factor a perfect square on the left side:
(x + 4)(x + 4) = 256

Calculate the square root of the right side: 16

Break this problem into two subproblems by setting 
(x + 4) equal to 16 and -16.

Subproblem 1

x + 4 = 16 Simplifying x + 4 = 16 Reorder the terms: 4 + x = 16 Solving 4 + x = 16 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 16 + -4 Combine like terms: 4 + -4 = 0 0 + x = 16 + -4 x = 16 + -4 Combine like terms: 16 + -4 = 12 x = 12 Simplifying x = 12

Subproblem 2

x + 4 = -16 Simplifying x + 4 = -16 Reorder the terms: 4 + x = -16 Solving 4 + x = -16 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -16 + -4 Combine like terms: 4 + -4 = 0 0 + x = -16 + -4 x = -16 + -4 Combine like terms: -16 + -4 = -20 x = -20 Simplifying x = -20

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12, -20}

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