0.5x^2+27=54-4.5*0.5x-3*0.5x

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Solution for 0.5x^2+27=54-4.5*0.5x-3*0.5x equation:


Simplifying
0.5x2 + 27 = 54 + -4.5 * 0.5x + -3 * 0.5x

Reorder the terms:
27 + 0.5x2 = 54 + -4.5 * 0.5x + -3 * 0.5x

Multiply -4.5 * 0.5
27 + 0.5x2 = 54 + -2.25x + -3 * 0.5x

Multiply -3 * 0.5
27 + 0.5x2 = 54 + -2.25x + -1.5x

Combine like terms: -2.25x + -1.5x = -3.75x
27 + 0.5x2 = 54 + -3.75x

Solving
27 + 0.5x2 = 54 + -3.75x

Solving for variable 'x'.

Reorder the terms:
27 + -54 + 3.75x + 0.5x2 = 54 + -3.75x + -54 + 3.75x

Combine like terms: 27 + -54 = -27
-27 + 3.75x + 0.5x2 = 54 + -3.75x + -54 + 3.75x

Reorder the terms:
-27 + 3.75x + 0.5x2 = 54 + -54 + -3.75x + 3.75x

Combine like terms: 54 + -54 = 0
-27 + 3.75x + 0.5x2 = 0 + -3.75x + 3.75x
-27 + 3.75x + 0.5x2 = -3.75x + 3.75x

Combine like terms: -3.75x + 3.75x = 0.00
-27 + 3.75x + 0.5x2 = 0.00

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

Divide each side by '0.5'.
-54 + 7.5x + x2 = 0.0

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 7.5x + 54 + x2 = 0.0 + 54

Reorder the terms:
-54 + 54 + 7.5x + x2 = 0.0 + 54

Combine like terms: -54 + 54 = 0
0 + 7.5x + x2 = 0.0 + 54
7.5x + x2 = 0.0 + 54

Combine like terms: 0.0 + 54 = 54
7.5x + x2 = 54

The x term is 7.5x.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5x + 14.0625 + x2 = 54 + 14.0625

Reorder the terms:
14.0625 + 7.5x + x2 = 54 + 14.0625

Combine like terms: 54 + 14.0625 = 68.0625
14.0625 + 7.5x + x2 = 68.0625

Factor a perfect square on the left side:
(x + 3.75)(x + 3.75) = 68.0625

Calculate the square root of the right side: 8.25

Break this problem into two subproblems by setting 
(x + 3.75) equal to 8.25 and -8.25.

Subproblem 1

x + 3.75 = 8.25 Simplifying x + 3.75 = 8.25 Reorder the terms: 3.75 + x = 8.25 Solving 3.75 + x = 8.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = 8.25 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 8.25 + -3.75 x = 8.25 + -3.75 Combine like terms: 8.25 + -3.75 = 4.5 x = 4.5 Simplifying x = 4.5

Subproblem 2

x + 3.75 = -8.25 Simplifying x + 3.75 = -8.25 Reorder the terms: 3.75 + x = -8.25 Solving 3.75 + x = -8.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = -8.25 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -8.25 + -3.75 x = -8.25 + -3.75 Combine like terms: -8.25 + -3.75 = -12 x = -12 Simplifying x = -12

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.5, -12}

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