42x^2-54x+1=0

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Solution for 42x^2-54x+1=0 equation:


Simplifying
42x2 + -54x + 1 = 0

Reorder the terms:
1 + -54x + 42x2 = 0

Solving
1 + -54x + 42x2 = 0

Solving for variable 'x'.

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

Divide each side by '42'.
0.02380952381 + -1.285714286x + x2 = 0

Move the constant term to the right:

Add '-0.02380952381' to each side of the equation.
0.02380952381 + -1.285714286x + -0.02380952381 + x2 = 0 + -0.02380952381

Reorder the terms:
0.02380952381 + -0.02380952381 + -1.285714286x + x2 = 0 + -0.02380952381

Combine like terms: 0.02380952381 + -0.02380952381 = 0.00000000000
0.00000000000 + -1.285714286x + x2 = 0 + -0.02380952381
-1.285714286x + x2 = 0 + -0.02380952381

Combine like terms: 0 + -0.02380952381 = -0.02380952381
-1.285714286x + x2 = -0.02380952381

The x term is -1.285714286x.  Take half its coefficient (-0.642857143).
Square it (0.4132653063) and add it to both sides.

Add '0.4132653063' to each side of the equation.
-1.285714286x + 0.4132653063 + x2 = -0.02380952381 + 0.4132653063

Reorder the terms:
0.4132653063 + -1.285714286x + x2 = -0.02380952381 + 0.4132653063

Combine like terms: -0.02380952381 + 0.4132653063 = 0.38945578249
0.4132653063 + -1.285714286x + x2 = 0.38945578249

Factor a perfect square on the left side:
(x + -0.642857143)(x + -0.642857143) = 0.38945578249

Calculate the square root of the right side: 0.624063925

Break this problem into two subproblems by setting 
(x + -0.642857143) equal to 0.624063925 and -0.624063925.

Subproblem 1

x + -0.642857143 = 0.624063925 Simplifying x + -0.642857143 = 0.624063925 Reorder the terms: -0.642857143 + x = 0.624063925 Solving -0.642857143 + x = 0.624063925 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + x = 0.624063925 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + x = 0.624063925 + 0.642857143 x = 0.624063925 + 0.642857143 Combine like terms: 0.624063925 + 0.642857143 = 1.266921068 x = 1.266921068 Simplifying x = 1.266921068

Subproblem 2

x + -0.642857143 = -0.624063925 Simplifying x + -0.642857143 = -0.624063925 Reorder the terms: -0.642857143 + x = -0.624063925 Solving -0.642857143 + x = -0.624063925 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + x = -0.624063925 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + x = -0.624063925 + 0.642857143 x = -0.624063925 + 0.642857143 Combine like terms: -0.624063925 + 0.642857143 = 0.018793218 x = 0.018793218 Simplifying x = 0.018793218

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.266921068, 0.018793218}

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