30x-x^2=112.5

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Solution for 30x-x^2=112.5 equation:


Simplifying
30x + -1x2 = 112.5

Solving
30x + -1x2 = 112.5

Solving for variable 'x'.

Reorder the terms:
-112.5 + 30x + -1x2 = 112.5 + -112.5

Combine like terms: 112.5 + -112.5 = 0.0
-112.5 + 30x + -1x2 = 0.0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
112.5 + -30x + x2 = 0.0

Move the constant term to the right:

Add '-112.5' to each side of the equation.
112.5 + -30x + -112.5 + x2 = 0.0 + -112.5

Reorder the terms:
112.5 + -112.5 + -30x + x2 = 0.0 + -112.5

Combine like terms: 112.5 + -112.5 = 0.0
0.0 + -30x + x2 = 0.0 + -112.5
-30x + x2 = 0.0 + -112.5

Combine like terms: 0.0 + -112.5 = -112.5
-30x + x2 = -112.5

The x term is -30x.  Take half its coefficient (-15).
Square it (225) and add it to both sides.

Add '225' to each side of the equation.
-30x + 225 + x2 = -112.5 + 225

Reorder the terms:
225 + -30x + x2 = -112.5 + 225

Combine like terms: -112.5 + 225 = 112.5
225 + -30x + x2 = 112.5

Factor a perfect square on the left side:
(x + -15)(x + -15) = 112.5

Calculate the square root of the right side: 10.606601718

Break this problem into two subproblems by setting 
(x + -15) equal to 10.606601718 and -10.606601718.

Subproblem 1

x + -15 = 10.606601718 Simplifying x + -15 = 10.606601718 Reorder the terms: -15 + x = 10.606601718 Solving -15 + x = 10.606601718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + x = 10.606601718 + 15 Combine like terms: -15 + 15 = 0 0 + x = 10.606601718 + 15 x = 10.606601718 + 15 Combine like terms: 10.606601718 + 15 = 25.606601718 x = 25.606601718 Simplifying x = 25.606601718

Subproblem 2

x + -15 = -10.606601718 Simplifying x + -15 = -10.606601718 Reorder the terms: -15 + x = -10.606601718 Solving -15 + x = -10.606601718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + x = -10.606601718 + 15 Combine like terms: -15 + 15 = 0 0 + x = -10.606601718 + 15 x = -10.606601718 + 15 Combine like terms: -10.606601718 + 15 = 4.393398282 x = 4.393398282 Simplifying x = 4.393398282

Solution

The solution to the problem is based on the solutions from the subproblems. x = {25.606601718, 4.393398282}

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