22.5=-10x^2+30x

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Solution for 22.5=-10x^2+30x equation:


Simplifying
22.5 = -10x2 + 30x

Reorder the terms:
22.5 = 30x + -10x2

Solving
22.5 = 30x + -10x2

Solving for variable 'x'.

Reorder the terms:
22.5 + -30x + 10x2 = 30x + -30x + -10x2 + 10x2

Combine like terms: 30x + -30x = 0
22.5 + -30x + 10x2 = 0 + -10x2 + 10x2
22.5 + -30x + 10x2 = -10x2 + 10x2

Combine like terms: -10x2 + 10x2 = 0
22.5 + -30x + 10x2 = 0

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

Divide each side by '10'.
2.25 + -3x + x2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + -3x + -2.25 + x2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + -3x + x2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + -3x + x2 = 0 + -2.25
-3x + x2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
-3x + x2 = -2.25

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 = -2.25 + 2.25

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

Combine like terms: -2.25 + 2.25 = 0.00
2.25 + -3x + x2 = 0.00

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 0.00

Calculate the square root of the right side: 0.00

Break this problem into two subproblems by setting 
(x + -1.5) equal to 0.00 and 0.00.

Subproblem 1

x + -1.5 = 0.00 Simplifying x + -1.5 = 0.00 Reorder the terms: -1.5 + x = 0.00 Solving -1.5 + x = 0.00 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 = 0.00 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.00 + 1.5 x = 0.00 + 1.5 Combine like terms: 0.00 + 1.5 = 1.5 x = 1.5 Simplifying x = 1.5

Subproblem 2

x + -1.5 = 0.00 Simplifying x + -1.5 = 0.00 Reorder the terms: -1.5 + x = 0.00 Solving -1.5 + x = 0.00 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 = 0.00 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.00 + 1.5 x = 0.00 + 1.5 Combine like terms: 0.00 + 1.5 = 1.5 x = 1.5 Simplifying x = 1.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5, 1.5}

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