23x^2-22x+1=0

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Solution for 23x^2-22x+1=0 equation:


Simplifying
23x2 + -22x + 1 = 0

Reorder the terms:
1 + -22x + 23x2 = 0

Solving
1 + -22x + 23x2 = 0

Solving for variable 'x'.

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

Divide each side by '23'.
0.04347826087 + -0.9565217391x + x2 = 0

Move the constant term to the right:

Add '-0.04347826087' to each side of the equation.
0.04347826087 + -0.9565217391x + -0.04347826087 + x2 = 0 + -0.04347826087

Reorder the terms:
0.04347826087 + -0.04347826087 + -0.9565217391x + x2 = 0 + -0.04347826087

Combine like terms: 0.04347826087 + -0.04347826087 = 0.00000000000
0.00000000000 + -0.9565217391x + x2 = 0 + -0.04347826087
-0.9565217391x + x2 = 0 + -0.04347826087

Combine like terms: 0 + -0.04347826087 = -0.04347826087
-0.9565217391x + x2 = -0.04347826087

The x term is -0.9565217391x.  Take half its coefficient (-0.4782608696).
Square it (0.2287334594) and add it to both sides.

Add '0.2287334594' to each side of the equation.
-0.9565217391x + 0.2287334594 + x2 = -0.04347826087 + 0.2287334594

Reorder the terms:
0.2287334594 + -0.9565217391x + x2 = -0.04347826087 + 0.2287334594

Combine like terms: -0.04347826087 + 0.2287334594 = 0.18525519853
0.2287334594 + -0.9565217391x + x2 = 0.18525519853

Factor a perfect square on the left side:
(x + -0.4782608696)(x + -0.4782608696) = 0.18525519853

Calculate the square root of the right side: 0.430412823

Break this problem into two subproblems by setting 
(x + -0.4782608696) equal to 0.430412823 and -0.430412823.

Subproblem 1

x + -0.4782608696 = 0.430412823 Simplifying x + -0.4782608696 = 0.430412823 Reorder the terms: -0.4782608696 + x = 0.430412823 Solving -0.4782608696 + x = 0.430412823 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4782608696' to each side of the equation. -0.4782608696 + 0.4782608696 + x = 0.430412823 + 0.4782608696 Combine like terms: -0.4782608696 + 0.4782608696 = 0.0000000000 0.0000000000 + x = 0.430412823 + 0.4782608696 x = 0.430412823 + 0.4782608696 Combine like terms: 0.430412823 + 0.4782608696 = 0.9086736926 x = 0.9086736926 Simplifying x = 0.9086736926

Subproblem 2

x + -0.4782608696 = -0.430412823 Simplifying x + -0.4782608696 = -0.430412823 Reorder the terms: -0.4782608696 + x = -0.430412823 Solving -0.4782608696 + x = -0.430412823 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4782608696' to each side of the equation. -0.4782608696 + 0.4782608696 + x = -0.430412823 + 0.4782608696 Combine like terms: -0.4782608696 + 0.4782608696 = 0.0000000000 0.0000000000 + x = -0.430412823 + 0.4782608696 x = -0.430412823 + 0.4782608696 Combine like terms: -0.430412823 + 0.4782608696 = 0.0478480466 x = 0.0478480466 Simplifying x = 0.0478480466

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.9086736926, 0.0478480466}

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