x(90-x)=18

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Solution for x(90-x)=18 equation:


Simplifying
x(90 + -1x) = 18
(90 * x + -1x * x) = 18
(90x + -1x2) = 18

Solving
90x + -1x2 = 18

Solving for variable 'x'.

Reorder the terms:
-18 + 90x + -1x2 = 18 + -18

Combine like terms: 18 + -18 = 0
-18 + 90x + -1x2 = 0

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

Divide each side by '-1'.
18 + -90x + x2 = 0

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -90x + -18 + x2 = 0 + -18

Reorder the terms:
18 + -18 + -90x + x2 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -90x + x2 = 0 + -18
-90x + x2 = 0 + -18

Combine like terms: 0 + -18 = -18
-90x + x2 = -18

The x term is -90x.  Take half its coefficient (-45).
Square it (2025) and add it to both sides.

Add '2025' to each side of the equation.
-90x + 2025 + x2 = -18 + 2025

Reorder the terms:
2025 + -90x + x2 = -18 + 2025

Combine like terms: -18 + 2025 = 2007
2025 + -90x + x2 = 2007

Factor a perfect square on the left side:
(x + -45)(x + -45) = 2007

Calculate the square root of the right side: 44.799553569

Break this problem into two subproblems by setting 
(x + -45) equal to 44.799553569 and -44.799553569.

Subproblem 1

x + -45 = 44.799553569 Simplifying x + -45 = 44.799553569 Reorder the terms: -45 + x = 44.799553569 Solving -45 + x = 44.799553569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = 44.799553569 + 45 Combine like terms: -45 + 45 = 0 0 + x = 44.799553569 + 45 x = 44.799553569 + 45 Combine like terms: 44.799553569 + 45 = 89.799553569 x = 89.799553569 Simplifying x = 89.799553569

Subproblem 2

x + -45 = -44.799553569 Simplifying x + -45 = -44.799553569 Reorder the terms: -45 + x = -44.799553569 Solving -45 + x = -44.799553569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = -44.799553569 + 45 Combine like terms: -45 + 45 = 0 0 + x = -44.799553569 + 45 x = -44.799553569 + 45 Combine like terms: -44.799553569 + 45 = 0.200446431 x = 0.200446431 Simplifying x = 0.200446431

Solution

The solution to the problem is based on the solutions from the subproblems. x = {89.799553569, 0.200446431}

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