21x^2-43x+19=0

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Solution for 21x^2-43x+19=0 equation:


Simplifying
21x2 + -43x + 19 = 0

Reorder the terms:
19 + -43x + 21x2 = 0

Solving
19 + -43x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
0.9047619048 + -2.047619048x + x2 = 0

Move the constant term to the right:

Add '-0.9047619048' to each side of the equation.
0.9047619048 + -2.047619048x + -0.9047619048 + x2 = 0 + -0.9047619048

Reorder the terms:
0.9047619048 + -0.9047619048 + -2.047619048x + x2 = 0 + -0.9047619048

Combine like terms: 0.9047619048 + -0.9047619048 = 0.0000000000
0.0000000000 + -2.047619048x + x2 = 0 + -0.9047619048
-2.047619048x + x2 = 0 + -0.9047619048

Combine like terms: 0 + -0.9047619048 = -0.9047619048
-2.047619048x + x2 = -0.9047619048

The x term is -2.047619048x.  Take half its coefficient (-1.023809524).
Square it (1.048185941) and add it to both sides.

Add '1.048185941' to each side of the equation.
-2.047619048x + 1.048185941 + x2 = -0.9047619048 + 1.048185941

Reorder the terms:
1.048185941 + -2.047619048x + x2 = -0.9047619048 + 1.048185941

Combine like terms: -0.9047619048 + 1.048185941 = 0.1434240362
1.048185941 + -2.047619048x + x2 = 0.1434240362

Factor a perfect square on the left side:
(x + -1.023809524)(x + -1.023809524) = 0.1434240362

Calculate the square root of the right side: 0.37871366

Break this problem into two subproblems by setting 
(x + -1.023809524) equal to 0.37871366 and -0.37871366.

Subproblem 1

x + -1.023809524 = 0.37871366 Simplifying x + -1.023809524 = 0.37871366 Reorder the terms: -1.023809524 + x = 0.37871366 Solving -1.023809524 + x = 0.37871366 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.023809524' to each side of the equation. -1.023809524 + 1.023809524 + x = 0.37871366 + 1.023809524 Combine like terms: -1.023809524 + 1.023809524 = 0.000000000 0.000000000 + x = 0.37871366 + 1.023809524 x = 0.37871366 + 1.023809524 Combine like terms: 0.37871366 + 1.023809524 = 1.402523184 x = 1.402523184 Simplifying x = 1.402523184

Subproblem 2

x + -1.023809524 = -0.37871366 Simplifying x + -1.023809524 = -0.37871366 Reorder the terms: -1.023809524 + x = -0.37871366 Solving -1.023809524 + x = -0.37871366 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.023809524' to each side of the equation. -1.023809524 + 1.023809524 + x = -0.37871366 + 1.023809524 Combine like terms: -1.023809524 + 1.023809524 = 0.000000000 0.000000000 + x = -0.37871366 + 1.023809524 x = -0.37871366 + 1.023809524 Combine like terms: -0.37871366 + 1.023809524 = 0.645095864 x = 0.645095864 Simplifying x = 0.645095864

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.402523184, 0.645095864}

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