50x-.5x^2=4x+10

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Solution for 50x-.5x^2=4x+10 equation:


Simplifying
50x + -0.5x2 = 4x + 10

Reorder the terms:
50x + -0.5x2 = 10 + 4x

Solving
50x + -0.5x2 = 10 + 4x

Solving for variable 'x'.

Reorder the terms:
-10 + 50x + -4x + -0.5x2 = 10 + 4x + -10 + -4x

Combine like terms: 50x + -4x = 46x
-10 + 46x + -0.5x2 = 10 + 4x + -10 + -4x

Reorder the terms:
-10 + 46x + -0.5x2 = 10 + -10 + 4x + -4x

Combine like terms: 10 + -10 = 0
-10 + 46x + -0.5x2 = 0 + 4x + -4x
-10 + 46x + -0.5x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-10 + 46x + -0.5x2 = 0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
20 + -92x + x2 = 0

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + -92x + -20 + x2 = 0 + -20

Reorder the terms:
20 + -20 + -92x + x2 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + -92x + x2 = 0 + -20
-92x + x2 = 0 + -20

Combine like terms: 0 + -20 = -20
-92x + x2 = -20

The x term is -92x.  Take half its coefficient (-46).
Square it (2116) and add it to both sides.

Add '2116' to each side of the equation.
-92x + 2116 + x2 = -20 + 2116

Reorder the terms:
2116 + -92x + x2 = -20 + 2116

Combine like terms: -20 + 2116 = 2096
2116 + -92x + x2 = 2096

Factor a perfect square on the left side:
(x + -46)(x + -46) = 2096

Calculate the square root of the right side: 45.782092569

Break this problem into two subproblems by setting 
(x + -46) equal to 45.782092569 and -45.782092569.

Subproblem 1

x + -46 = 45.782092569 Simplifying x + -46 = 45.782092569 Reorder the terms: -46 + x = 45.782092569 Solving -46 + x = 45.782092569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '46' to each side of the equation. -46 + 46 + x = 45.782092569 + 46 Combine like terms: -46 + 46 = 0 0 + x = 45.782092569 + 46 x = 45.782092569 + 46 Combine like terms: 45.782092569 + 46 = 91.782092569 x = 91.782092569 Simplifying x = 91.782092569

Subproblem 2

x + -46 = -45.782092569 Simplifying x + -46 = -45.782092569 Reorder the terms: -46 + x = -45.782092569 Solving -46 + x = -45.782092569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '46' to each side of the equation. -46 + 46 + x = -45.782092569 + 46 Combine like terms: -46 + 46 = 0 0 + x = -45.782092569 + 46 x = -45.782092569 + 46 Combine like terms: -45.782092569 + 46 = 0.217907431 x = 0.217907431 Simplifying x = 0.217907431

Solution

The solution to the problem is based on the solutions from the subproblems. x = {91.782092569, 0.217907431}

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