120x-x^2=77

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Solution for 120x-x^2=77 equation:


Simplifying
120x + -1x2 = 77

Solving
120x + -1x2 = 77

Solving for variable 'x'.

Reorder the terms:
-77 + 120x + -1x2 = 77 + -77

Combine like terms: 77 + -77 = 0
-77 + 120x + -1x2 = 0

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

Divide each side by '-1'.
77 + -120x + x2 = 0

Move the constant term to the right:

Add '-77' to each side of the equation.
77 + -120x + -77 + x2 = 0 + -77

Reorder the terms:
77 + -77 + -120x + x2 = 0 + -77

Combine like terms: 77 + -77 = 0
0 + -120x + x2 = 0 + -77
-120x + x2 = 0 + -77

Combine like terms: 0 + -77 = -77
-120x + x2 = -77

The x term is -120x.  Take half its coefficient (-60).
Square it (3600) and add it to both sides.

Add '3600' to each side of the equation.
-120x + 3600 + x2 = -77 + 3600

Reorder the terms:
3600 + -120x + x2 = -77 + 3600

Combine like terms: -77 + 3600 = 3523
3600 + -120x + x2 = 3523

Factor a perfect square on the left side:
(x + -60)(x + -60) = 3523

Calculate the square root of the right side: 59.354865007

Break this problem into two subproblems by setting 
(x + -60) equal to 59.354865007 and -59.354865007.

Subproblem 1

x + -60 = 59.354865007 Simplifying x + -60 = 59.354865007 Reorder the terms: -60 + x = 59.354865007 Solving -60 + x = 59.354865007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + x = 59.354865007 + 60 Combine like terms: -60 + 60 = 0 0 + x = 59.354865007 + 60 x = 59.354865007 + 60 Combine like terms: 59.354865007 + 60 = 119.354865007 x = 119.354865007 Simplifying x = 119.354865007

Subproblem 2

x + -60 = -59.354865007 Simplifying x + -60 = -59.354865007 Reorder the terms: -60 + x = -59.354865007 Solving -60 + x = -59.354865007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + x = -59.354865007 + 60 Combine like terms: -60 + 60 = 0 0 + x = -59.354865007 + 60 x = -59.354865007 + 60 Combine like terms: -59.354865007 + 60 = 0.645134993 x = 0.645134993 Simplifying x = 0.645134993

Solution

The solution to the problem is based on the solutions from the subproblems. x = {119.354865007, 0.645134993}

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