9x^2-37x+31=0

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Solution for 9x^2-37x+31=0 equation:


Simplifying
9x2 + -37x + 31 = 0

Reorder the terms:
31 + -37x + 9x2 = 0

Solving
31 + -37x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
3.444444444 + -4.111111111x + x2 = 0

Move the constant term to the right:

Add '-3.444444444' to each side of the equation.
3.444444444 + -4.111111111x + -3.444444444 + x2 = 0 + -3.444444444

Reorder the terms:
3.444444444 + -3.444444444 + -4.111111111x + x2 = 0 + -3.444444444

Combine like terms: 3.444444444 + -3.444444444 = 0.000000000
0.000000000 + -4.111111111x + x2 = 0 + -3.444444444
-4.111111111x + x2 = 0 + -3.444444444

Combine like terms: 0 + -3.444444444 = -3.444444444
-4.111111111x + x2 = -3.444444444

The x term is -4.111111111x.  Take half its coefficient (-2.055555556).
Square it (4.225308644) and add it to both sides.

Add '4.225308644' to each side of the equation.
-4.111111111x + 4.225308644 + x2 = -3.444444444 + 4.225308644

Reorder the terms:
4.225308644 + -4.111111111x + x2 = -3.444444444 + 4.225308644

Combine like terms: -3.444444444 + 4.225308644 = 0.7808642
4.225308644 + -4.111111111x + x2 = 0.7808642

Factor a perfect square on the left side:
(x + -2.055555556)(x + -2.055555556) = 0.7808642

Calculate the square root of the right side: 0.883665208

Break this problem into two subproblems by setting 
(x + -2.055555556) equal to 0.883665208 and -0.883665208.

Subproblem 1

x + -2.055555556 = 0.883665208 Simplifying x + -2.055555556 = 0.883665208 Reorder the terms: -2.055555556 + x = 0.883665208 Solving -2.055555556 + x = 0.883665208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.055555556' to each side of the equation. -2.055555556 + 2.055555556 + x = 0.883665208 + 2.055555556 Combine like terms: -2.055555556 + 2.055555556 = 0.000000000 0.000000000 + x = 0.883665208 + 2.055555556 x = 0.883665208 + 2.055555556 Combine like terms: 0.883665208 + 2.055555556 = 2.939220764 x = 2.939220764 Simplifying x = 2.939220764

Subproblem 2

x + -2.055555556 = -0.883665208 Simplifying x + -2.055555556 = -0.883665208 Reorder the terms: -2.055555556 + x = -0.883665208 Solving -2.055555556 + x = -0.883665208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.055555556' to each side of the equation. -2.055555556 + 2.055555556 + x = -0.883665208 + 2.055555556 Combine like terms: -2.055555556 + 2.055555556 = 0.000000000 0.000000000 + x = -0.883665208 + 2.055555556 x = -0.883665208 + 2.055555556 Combine like terms: -0.883665208 + 2.055555556 = 1.171890348 x = 1.171890348 Simplifying x = 1.171890348

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.939220764, 1.171890348}

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