180x^2-220x+21=0

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Solution for 180x^2-220x+21=0 equation:


Simplifying
180x2 + -220x + 21 = 0

Reorder the terms:
21 + -220x + 180x2 = 0

Solving
21 + -220x + 180x2 = 0

Solving for variable 'x'.

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

Divide each side by '180'.
0.1166666667 + -1.222222222x + x2 = 0

Move the constant term to the right:

Add '-0.1166666667' to each side of the equation.
0.1166666667 + -1.222222222x + -0.1166666667 + x2 = 0 + -0.1166666667

Reorder the terms:
0.1166666667 + -0.1166666667 + -1.222222222x + x2 = 0 + -0.1166666667

Combine like terms: 0.1166666667 + -0.1166666667 = 0.0000000000
0.0000000000 + -1.222222222x + x2 = 0 + -0.1166666667
-1.222222222x + x2 = 0 + -0.1166666667

Combine like terms: 0 + -0.1166666667 = -0.1166666667
-1.222222222x + x2 = -0.1166666667

The x term is -1.222222222x.  Take half its coefficient (-0.611111111).
Square it (0.3734567900) and add it to both sides.

Add '0.3734567900' to each side of the equation.
-1.222222222x + 0.3734567900 + x2 = -0.1166666667 + 0.3734567900

Reorder the terms:
0.3734567900 + -1.222222222x + x2 = -0.1166666667 + 0.3734567900

Combine like terms: -0.1166666667 + 0.3734567900 = 0.2567901233
0.3734567900 + -1.222222222x + x2 = 0.2567901233

Factor a perfect square on the left side:
(x + -0.611111111)(x + -0.611111111) = 0.2567901233

Calculate the square root of the right side: 0.506744633

Break this problem into two subproblems by setting 
(x + -0.611111111) equal to 0.506744633 and -0.506744633.

Subproblem 1

x + -0.611111111 = 0.506744633 Simplifying x + -0.611111111 = 0.506744633 Reorder the terms: -0.611111111 + x = 0.506744633 Solving -0.611111111 + x = 0.506744633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + x = 0.506744633 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + x = 0.506744633 + 0.611111111 x = 0.506744633 + 0.611111111 Combine like terms: 0.506744633 + 0.611111111 = 1.117855744 x = 1.117855744 Simplifying x = 1.117855744

Subproblem 2

x + -0.611111111 = -0.506744633 Simplifying x + -0.611111111 = -0.506744633 Reorder the terms: -0.611111111 + x = -0.506744633 Solving -0.611111111 + x = -0.506744633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + x = -0.506744633 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + x = -0.506744633 + 0.611111111 x = -0.506744633 + 0.611111111 Combine like terms: -0.506744633 + 0.611111111 = 0.104366478 x = 0.104366478 Simplifying x = 0.104366478

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.117855744, 0.104366478}

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