0=-0.25x^2+6x-27

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Solution for 0=-0.25x^2+6x-27 equation:


Simplifying
0 = -0.25x2 + 6x + -27

Reorder the terms:
0 = -27 + 6x + -0.25x2

Solving
0 = -27 + 6x + -0.25x2

Solving for variable 'x'.

Combine like terms: 0 + 27 = 27
27 + -6x + 0.25x2 = -27 + 6x + -0.25x2 + 27 + -6x + 0.25x2

Reorder the terms:
27 + -6x + 0.25x2 = -27 + 27 + 6x + -6x + -0.25x2 + 0.25x2

Combine like terms: -27 + 27 = 0
27 + -6x + 0.25x2 = 0 + 6x + -6x + -0.25x2 + 0.25x2
27 + -6x + 0.25x2 = 6x + -6x + -0.25x2 + 0.25x2

Combine like terms: 6x + -6x = 0
27 + -6x + 0.25x2 = 0 + -0.25x2 + 0.25x2
27 + -6x + 0.25x2 = -0.25x2 + 0.25x2

Combine like terms: -0.25x2 + 0.25x2 = 0.00
27 + -6x + 0.25x2 = 0.00

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

Divide each side by '0.25'.
108 + -24x + x2 = 0

Move the constant term to the right:

Add '-108' to each side of the equation.
108 + -24x + -108 + x2 = 0 + -108

Reorder the terms:
108 + -108 + -24x + x2 = 0 + -108

Combine like terms: 108 + -108 = 0
0 + -24x + x2 = 0 + -108
-24x + x2 = 0 + -108

Combine like terms: 0 + -108 = -108
-24x + x2 = -108

The x term is -24x.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24x + 144 + x2 = -108 + 144

Reorder the terms:
144 + -24x + x2 = -108 + 144

Combine like terms: -108 + 144 = 36
144 + -24x + x2 = 36

Factor a perfect square on the left side:
(x + -12)(x + -12) = 36

Calculate the square root of the right side: 6

Break this problem into two subproblems by setting 
(x + -12) equal to 6 and -6.

Subproblem 1

x + -12 = 6 Simplifying x + -12 = 6 Reorder the terms: -12 + x = 6 Solving -12 + x = 6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = 6 + 12 Combine like terms: -12 + 12 = 0 0 + x = 6 + 12 x = 6 + 12 Combine like terms: 6 + 12 = 18 x = 18 Simplifying x = 18

Subproblem 2

x + -12 = -6 Simplifying x + -12 = -6 Reorder the terms: -12 + x = -6 Solving -12 + x = -6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = -6 + 12 Combine like terms: -12 + 12 = 0 0 + x = -6 + 12 x = -6 + 12 Combine like terms: -6 + 12 = 6 x = 6 Simplifying x = 6

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18, 6}

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