100x-2x^2=100+x

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Solution for 100x-2x^2=100+x equation:


Simplifying
100x + -2x2 = 100 + x

Solving
100x + -2x2 = 100 + x

Solving for variable 'x'.

Reorder the terms:
-100 + 100x + -1x + -2x2 = 100 + x + -100 + -1x

Combine like terms: 100x + -1x = 99x
-100 + 99x + -2x2 = 100 + x + -100 + -1x

Reorder the terms:
-100 + 99x + -2x2 = 100 + -100 + x + -1x

Combine like terms: 100 + -100 = 0
-100 + 99x + -2x2 = 0 + x + -1x
-100 + 99x + -2x2 = x + -1x

Combine like terms: x + -1x = 0
-100 + 99x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
50 + -49.5x + x2 = 0

Move the constant term to the right:

Add '-50' to each side of the equation.
50 + -49.5x + -50 + x2 = 0 + -50

Reorder the terms:
50 + -50 + -49.5x + x2 = 0 + -50

Combine like terms: 50 + -50 = 0
0 + -49.5x + x2 = 0 + -50
-49.5x + x2 = 0 + -50

Combine like terms: 0 + -50 = -50
-49.5x + x2 = -50

The x term is -49.5x.  Take half its coefficient (-24.75).
Square it (612.5625) and add it to both sides.

Add '612.5625' to each side of the equation.
-49.5x + 612.5625 + x2 = -50 + 612.5625

Reorder the terms:
612.5625 + -49.5x + x2 = -50 + 612.5625

Combine like terms: -50 + 612.5625 = 562.5625
612.5625 + -49.5x + x2 = 562.5625

Factor a perfect square on the left side:
(x + -24.75)(x + -24.75) = 562.5625

Calculate the square root of the right side: 23.71840003

Break this problem into two subproblems by setting 
(x + -24.75) equal to 23.71840003 and -23.71840003.

Subproblem 1

x + -24.75 = 23.71840003 Simplifying x + -24.75 = 23.71840003 Reorder the terms: -24.75 + x = 23.71840003 Solving -24.75 + x = 23.71840003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.75' to each side of the equation. -24.75 + 24.75 + x = 23.71840003 + 24.75 Combine like terms: -24.75 + 24.75 = 0.00 0.00 + x = 23.71840003 + 24.75 x = 23.71840003 + 24.75 Combine like terms: 23.71840003 + 24.75 = 48.46840003 x = 48.46840003 Simplifying x = 48.46840003

Subproblem 2

x + -24.75 = -23.71840003 Simplifying x + -24.75 = -23.71840003 Reorder the terms: -24.75 + x = -23.71840003 Solving -24.75 + x = -23.71840003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.75' to each side of the equation. -24.75 + 24.75 + x = -23.71840003 + 24.75 Combine like terms: -24.75 + 24.75 = 0.00 0.00 + x = -23.71840003 + 24.75 x = -23.71840003 + 24.75 Combine like terms: -23.71840003 + 24.75 = 1.03159997 x = 1.03159997 Simplifying x = 1.03159997

Solution

The solution to the problem is based on the solutions from the subproblems. x = {48.46840003, 1.03159997}

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