0.5x^2+1.5x-54=0

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Solution for 0.5x^2+1.5x-54=0 equation:


Simplifying
0.5x2 + 1.5x + -54 = 0

Reorder the terms:
-54 + 1.5x + 0.5x2 = 0

Solving
-54 + 1.5x + 0.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.5'.
-108 + 3x + x2 = 0

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + 3x + 108 + x2 = 0 + 108

Reorder the terms:
-108 + 108 + 3x + x2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + 3x + x2 = 0 + 108
3x + x2 = 0 + 108

Combine like terms: 0 + 108 = 108
3x + x2 = 108

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 108 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 108 + 2.25

Combine like terms: 108 + 2.25 = 110.25
2.25 + 3x + x2 = 110.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 110.25

Calculate the square root of the right side: 10.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 10.5 and -10.5.

Subproblem 1

x + 1.5 = 10.5 Simplifying x + 1.5 = 10.5 Reorder the terms: 1.5 + x = 10.5 Solving 1.5 + x = 10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 10.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 10.5 + -1.5 x = 10.5 + -1.5 Combine like terms: 10.5 + -1.5 = 9 x = 9 Simplifying x = 9

Subproblem 2

x + 1.5 = -10.5 Simplifying x + 1.5 = -10.5 Reorder the terms: 1.5 + x = -10.5 Solving 1.5 + x = -10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -10.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -10.5 + -1.5 x = -10.5 + -1.5 Combine like terms: -10.5 + -1.5 = -12 x = -12 Simplifying x = -12

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9, -12}

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