-1.5Q^2+24Q-22.5=0

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Solution for -1.5Q^2+24Q-22.5=0 equation:


Simplifying
-1.5Q2 + 24Q + -22.5 = 0

Reorder the terms:
-22.5 + 24Q + -1.5Q2 = 0

Solving
-22.5 + 24Q + -1.5Q2 = 0

Solving for variable 'Q'.

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

Divide each side by '-1.5'.
15 + -16Q + Q2 = 0

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -16Q + -15 + Q2 = 0 + -15

Reorder the terms:
15 + -15 + -16Q + Q2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -16Q + Q2 = 0 + -15
-16Q + Q2 = 0 + -15

Combine like terms: 0 + -15 = -15
-16Q + Q2 = -15

The Q term is -16Q.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16Q + 64 + Q2 = -15 + 64

Reorder the terms:
64 + -16Q + Q2 = -15 + 64

Combine like terms: -15 + 64 = 49
64 + -16Q + Q2 = 49

Factor a perfect square on the left side:
(Q + -8)(Q + -8) = 49

Calculate the square root of the right side: 7

Break this problem into two subproblems by setting 
(Q + -8) equal to 7 and -7.

Subproblem 1

Q + -8 = 7 Simplifying Q + -8 = 7 Reorder the terms: -8 + Q = 7 Solving -8 + Q = 7 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + Q = 7 + 8 Combine like terms: -8 + 8 = 0 0 + Q = 7 + 8 Q = 7 + 8 Combine like terms: 7 + 8 = 15 Q = 15 Simplifying Q = 15

Subproblem 2

Q + -8 = -7 Simplifying Q + -8 = -7 Reorder the terms: -8 + Q = -7 Solving -8 + Q = -7 Solving for variable 'Q'. Move all terms containing Q to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + Q = -7 + 8 Combine like terms: -8 + 8 = 0 0 + Q = -7 + 8 Q = -7 + 8 Combine like terms: -7 + 8 = 1 Q = 1 Simplifying Q = 1

Solution

The solution to the problem is based on the solutions from the subproblems. Q = {15, 1}

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