-1.5x^2+33x-75=52.50

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Solution for -1.5x^2+33x-75=52.50 equation:


Simplifying
-1.5x2 + 33x + -75 = 52.50

Reorder the terms:
-75 + 33x + -1.5x2 = 52.50

Solving
-75 + 33x + -1.5x2 = 52.50

Solving for variable 'x'.

Reorder the terms:
-75 + -52.50 + 33x + -1.5x2 = 52.50 + -52.50

Combine like terms: -75 + -52.50 = -127.5
-127.5 + 33x + -1.5x2 = 52.50 + -52.50

Combine like terms: 52.50 + -52.50 = 0.00
-127.5 + 33x + -1.5x2 = 0.00

Begin completing the square.  Divide all terms by
-1.5 the coefficient of the squared term: 

Divide each side by '-1.5'.
85 + -22x + x2 = 0.0

Move the constant term to the right:

Add '-85' to each side of the equation.
85 + -22x + -85 + x2 = 0.0 + -85

Reorder the terms:
85 + -85 + -22x + x2 = 0.0 + -85

Combine like terms: 85 + -85 = 0
0 + -22x + x2 = 0.0 + -85
-22x + x2 = 0.0 + -85

Combine like terms: 0.0 + -85 = -85
-22x + x2 = -85

The x term is -22x.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22x + 121 + x2 = -85 + 121

Reorder the terms:
121 + -22x + x2 = -85 + 121

Combine like terms: -85 + 121 = 36
121 + -22x + x2 = 36

Factor a perfect square on the left side:
(x + -11)(x + -11) = 36

Calculate the square root of the right side: 6

Break this problem into two subproblems by setting 
(x + -11) equal to 6 and -6.

Subproblem 1

x + -11 = 6 Simplifying x + -11 = 6 Reorder the terms: -11 + x = 6 Solving -11 + x = 6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x = 6 + 11 Combine like terms: -11 + 11 = 0 0 + x = 6 + 11 x = 6 + 11 Combine like terms: 6 + 11 = 17 x = 17 Simplifying x = 17

Subproblem 2

x + -11 = -6 Simplifying x + -11 = -6 Reorder the terms: -11 + x = -6 Solving -11 + x = -6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x = -6 + 11 Combine like terms: -11 + 11 = 0 0 + x = -6 + 11 x = -6 + 11 Combine like terms: -6 + 11 = 5 x = 5 Simplifying x = 5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {17, 5}

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