21x^2-69x+20=0

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Solution for 21x^2-69x+20=0 equation:


Simplifying
21x2 + -69x + 20 = 0

Reorder the terms:
20 + -69x + 21x2 = 0

Solving
20 + -69x + 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'.
0.9523809524 + -3.285714286x + x2 = 0

Move the constant term to the right:

Add '-0.9523809524' to each side of the equation.
0.9523809524 + -3.285714286x + -0.9523809524 + x2 = 0 + -0.9523809524

Reorder the terms:
0.9523809524 + -0.9523809524 + -3.285714286x + x2 = 0 + -0.9523809524

Combine like terms: 0.9523809524 + -0.9523809524 = 0.0000000000
0.0000000000 + -3.285714286x + x2 = 0 + -0.9523809524
-3.285714286x + x2 = 0 + -0.9523809524

Combine like terms: 0 + -0.9523809524 = -0.9523809524
-3.285714286x + x2 = -0.9523809524

The x term is -3.285714286x.  Take half its coefficient (-1.642857143).
Square it (2.698979592) and add it to both sides.

Add '2.698979592' to each side of the equation.
-3.285714286x + 2.698979592 + x2 = -0.9523809524 + 2.698979592

Reorder the terms:
2.698979592 + -3.285714286x + x2 = -0.9523809524 + 2.698979592

Combine like terms: -0.9523809524 + 2.698979592 = 1.7465986396
2.698979592 + -3.285714286x + x2 = 1.7465986396

Factor a perfect square on the left side:
(x + -1.642857143)(x + -1.642857143) = 1.7465986396

Calculate the square root of the right side: 1.321589437

Break this problem into two subproblems by setting 
(x + -1.642857143) equal to 1.321589437 and -1.321589437.

Subproblem 1

x + -1.642857143 = 1.321589437 Simplifying x + -1.642857143 = 1.321589437 Reorder the terms: -1.642857143 + x = 1.321589437 Solving -1.642857143 + x = 1.321589437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.642857143' to each side of the equation. -1.642857143 + 1.642857143 + x = 1.321589437 + 1.642857143 Combine like terms: -1.642857143 + 1.642857143 = 0.000000000 0.000000000 + x = 1.321589437 + 1.642857143 x = 1.321589437 + 1.642857143 Combine like terms: 1.321589437 + 1.642857143 = 2.96444658 x = 2.96444658 Simplifying x = 2.96444658

Subproblem 2

x + -1.642857143 = -1.321589437 Simplifying x + -1.642857143 = -1.321589437 Reorder the terms: -1.642857143 + x = -1.321589437 Solving -1.642857143 + x = -1.321589437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.642857143' to each side of the equation. -1.642857143 + 1.642857143 + x = -1.321589437 + 1.642857143 Combine like terms: -1.642857143 + 1.642857143 = 0.000000000 0.000000000 + x = -1.321589437 + 1.642857143 x = -1.321589437 + 1.642857143 Combine like terms: -1.321589437 + 1.642857143 = 0.321267706 x = 0.321267706 Simplifying x = 0.321267706

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.96444658, 0.321267706}

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