3x^2-21x+31=0

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Solution for 3x^2-21x+31=0 equation:


Simplifying
3x2 + -21x + 31 = 0

Reorder the terms:
31 + -21x + 3x2 = 0

Solving
31 + -21x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
10.33333333 + -7x + x2 = 0

Move the constant term to the right:

Add '-10.33333333' to each side of the equation.
10.33333333 + -7x + -10.33333333 + x2 = 0 + -10.33333333

Reorder the terms:
10.33333333 + -10.33333333 + -7x + x2 = 0 + -10.33333333

Combine like terms: 10.33333333 + -10.33333333 = 0.00000000
0.00000000 + -7x + x2 = 0 + -10.33333333
-7x + x2 = 0 + -10.33333333

Combine like terms: 0 + -10.33333333 = -10.33333333
-7x + x2 = -10.33333333

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -10.33333333 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -10.33333333 + 12.25

Combine like terms: -10.33333333 + 12.25 = 1.91666667
12.25 + -7x + x2 = 1.91666667

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 1.91666667

Calculate the square root of the right side: 1.384437312

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1.384437312 and -1.384437312.

Subproblem 1

x + -3.5 = 1.384437312 Simplifying x + -3.5 = 1.384437312 Reorder the terms: -3.5 + x = 1.384437312 Solving -3.5 + x = 1.384437312 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 1.384437312 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 1.384437312 + 3.5 x = 1.384437312 + 3.5 Combine like terms: 1.384437312 + 3.5 = 4.884437312 x = 4.884437312 Simplifying x = 4.884437312

Subproblem 2

x + -3.5 = -1.384437312 Simplifying x + -3.5 = -1.384437312 Reorder the terms: -3.5 + x = -1.384437312 Solving -3.5 + x = -1.384437312 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -1.384437312 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -1.384437312 + 3.5 x = -1.384437312 + 3.5 Combine like terms: -1.384437312 + 3.5 = 2.115562688 x = 2.115562688 Simplifying x = 2.115562688

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.884437312, 2.115562688}

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