10x-x^2=17.5

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Solution for 10x-x^2=17.5 equation:


Simplifying
10x + -1x2 = 17.5

Solving
10x + -1x2 = 17.5

Solving for variable 'x'.

Reorder the terms:
-17.5 + 10x + -1x2 = 17.5 + -17.5

Combine like terms: 17.5 + -17.5 = 0.0
-17.5 + 10x + -1x2 = 0.0

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

Divide each side by '-1'.
17.5 + -10x + x2 = 0.0

Move the constant term to the right:

Add '-17.5' to each side of the equation.
17.5 + -10x + -17.5 + x2 = 0.0 + -17.5

Reorder the terms:
17.5 + -17.5 + -10x + x2 = 0.0 + -17.5

Combine like terms: 17.5 + -17.5 = 0.0
0.0 + -10x + x2 = 0.0 + -17.5
-10x + x2 = 0.0 + -17.5

Combine like terms: 0.0 + -17.5 = -17.5
-10x + x2 = -17.5

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -17.5 + 25

Reorder the terms:
25 + -10x + x2 = -17.5 + 25

Combine like terms: -17.5 + 25 = 7.5
25 + -10x + x2 = 7.5

Factor a perfect square on the left side:
(x + -5)(x + -5) = 7.5

Calculate the square root of the right side: 2.738612788

Break this problem into two subproblems by setting 
(x + -5) equal to 2.738612788 and -2.738612788.

Subproblem 1

x + -5 = 2.738612788 Simplifying x + -5 = 2.738612788 Reorder the terms: -5 + x = 2.738612788 Solving -5 + x = 2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 2.738612788 + 5 Combine like terms: -5 + 5 = 0 0 + x = 2.738612788 + 5 x = 2.738612788 + 5 Combine like terms: 2.738612788 + 5 = 7.738612788 x = 7.738612788 Simplifying x = 7.738612788

Subproblem 2

x + -5 = -2.738612788 Simplifying x + -5 = -2.738612788 Reorder the terms: -5 + x = -2.738612788 Solving -5 + x = -2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -2.738612788 + 5 Combine like terms: -5 + 5 = 0 0 + x = -2.738612788 + 5 x = -2.738612788 + 5 Combine like terms: -2.738612788 + 5 = 2.261387212 x = 2.261387212 Simplifying x = 2.261387212

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.738612788, 2.261387212}

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