7x^2-21x+5=0

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Solution for 7x^2-21x+5=0 equation:


Simplifying
7x2 + -21x + 5 = 0

Reorder the terms:
5 + -21x + 7x2 = 0

Solving
5 + -21x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
0.7142857143 + -3x + x2 = 0

Move the constant term to the right:

Add '-0.7142857143' to each side of the equation.
0.7142857143 + -3x + -0.7142857143 + x2 = 0 + -0.7142857143

Reorder the terms:
0.7142857143 + -0.7142857143 + -3x + x2 = 0 + -0.7142857143

Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000
0.0000000000 + -3x + x2 = 0 + -0.7142857143
-3x + x2 = 0 + -0.7142857143

Combine like terms: 0 + -0.7142857143 = -0.7142857143
-3x + x2 = -0.7142857143

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = -0.7142857143 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -0.7142857143 + 2.25

Combine like terms: -0.7142857143 + 2.25 = 1.5357142857
2.25 + -3x + x2 = 1.5357142857

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 1.5357142857

Calculate the square root of the right side: 1.239239398

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.239239398 and -1.239239398.

Subproblem 1

x + -1.5 = 1.239239398 Simplifying x + -1.5 = 1.239239398 Reorder the terms: -1.5 + x = 1.239239398 Solving -1.5 + x = 1.239239398 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.239239398 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.239239398 + 1.5 x = 1.239239398 + 1.5 Combine like terms: 1.239239398 + 1.5 = 2.739239398 x = 2.739239398 Simplifying x = 2.739239398

Subproblem 2

x + -1.5 = -1.239239398 Simplifying x + -1.5 = -1.239239398 Reorder the terms: -1.5 + x = -1.239239398 Solving -1.5 + x = -1.239239398 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.239239398 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.239239398 + 1.5 x = -1.239239398 + 1.5 Combine like terms: -1.239239398 + 1.5 = 0.260760602 x = 0.260760602 Simplifying x = 0.260760602

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.739239398, 0.260760602}

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