x^2-42x+180=0

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Solution for x^2-42x+180=0 equation:


Simplifying
x2 + -42x + 180 = 0

Reorder the terms:
180 + -42x + x2 = 0

Solving
180 + -42x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-180' to each side of the equation.
180 + -42x + -180 + x2 = 0 + -180

Reorder the terms:
180 + -180 + -42x + x2 = 0 + -180

Combine like terms: 180 + -180 = 0
0 + -42x + x2 = 0 + -180
-42x + x2 = 0 + -180

Combine like terms: 0 + -180 = -180
-42x + x2 = -180

The x term is -42x.  Take half its coefficient (-21).
Square it (441) and add it to both sides.

Add '441' to each side of the equation.
-42x + 441 + x2 = -180 + 441

Reorder the terms:
441 + -42x + x2 = -180 + 441

Combine like terms: -180 + 441 = 261
441 + -42x + x2 = 261

Factor a perfect square on the left side:
(x + -21)(x + -21) = 261

Calculate the square root of the right side: 16.155494421

Break this problem into two subproblems by setting 
(x + -21) equal to 16.155494421 and -16.155494421.

Subproblem 1

x + -21 = 16.155494421 Simplifying x + -21 = 16.155494421 Reorder the terms: -21 + x = 16.155494421 Solving -21 + x = 16.155494421 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21' to each side of the equation. -21 + 21 + x = 16.155494421 + 21 Combine like terms: -21 + 21 = 0 0 + x = 16.155494421 + 21 x = 16.155494421 + 21 Combine like terms: 16.155494421 + 21 = 37.155494421 x = 37.155494421 Simplifying x = 37.155494421

Subproblem 2

x + -21 = -16.155494421 Simplifying x + -21 = -16.155494421 Reorder the terms: -21 + x = -16.155494421 Solving -21 + x = -16.155494421 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21' to each side of the equation. -21 + 21 + x = -16.155494421 + 21 Combine like terms: -21 + 21 = 0 0 + x = -16.155494421 + 21 x = -16.155494421 + 21 Combine like terms: -16.155494421 + 21 = 4.844505579 x = 4.844505579 Simplifying x = 4.844505579

Solution

The solution to the problem is based on the solutions from the subproblems. x = {37.155494421, 4.844505579}

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