9x^2-31x=51

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Solution for 9x^2-31x=51 equation:


Simplifying
9x2 + -31x = 51

Reorder the terms:
-31x + 9x2 = 51

Solving
-31x + 9x2 = 51

Solving for variable 'x'.

Reorder the terms:
-51 + -31x + 9x2 = 51 + -51

Combine like terms: 51 + -51 = 0
-51 + -31x + 9x2 = 0

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

Divide each side by '9'.
-5.666666667 + -3.444444444x + x2 = 0

Move the constant term to the right:

Add '5.666666667' to each side of the equation.
-5.666666667 + -3.444444444x + 5.666666667 + x2 = 0 + 5.666666667

Reorder the terms:
-5.666666667 + 5.666666667 + -3.444444444x + x2 = 0 + 5.666666667

Combine like terms: -5.666666667 + 5.666666667 = 0.000000000
0.000000000 + -3.444444444x + x2 = 0 + 5.666666667
-3.444444444x + x2 = 0 + 5.666666667

Combine like terms: 0 + 5.666666667 = 5.666666667
-3.444444444x + x2 = 5.666666667

The x term is -3.444444444x.  Take half its coefficient (-1.722222222).
Square it (2.966049382) and add it to both sides.

Add '2.966049382' to each side of the equation.
-3.444444444x + 2.966049382 + x2 = 5.666666667 + 2.966049382

Reorder the terms:
2.966049382 + -3.444444444x + x2 = 5.666666667 + 2.966049382

Combine like terms: 5.666666667 + 2.966049382 = 8.632716049
2.966049382 + -3.444444444x + x2 = 8.632716049

Factor a perfect square on the left side:
(x + -1.722222222)(x + -1.722222222) = 8.632716049

Calculate the square root of the right side: 2.938148405

Break this problem into two subproblems by setting 
(x + -1.722222222) equal to 2.938148405 and -2.938148405.

Subproblem 1

x + -1.722222222 = 2.938148405 Simplifying x + -1.722222222 = 2.938148405 Reorder the terms: -1.722222222 + x = 2.938148405 Solving -1.722222222 + x = 2.938148405 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.722222222' to each side of the equation. -1.722222222 + 1.722222222 + x = 2.938148405 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x = 2.938148405 + 1.722222222 x = 2.938148405 + 1.722222222 Combine like terms: 2.938148405 + 1.722222222 = 4.660370627 x = 4.660370627 Simplifying x = 4.660370627

Subproblem 2

x + -1.722222222 = -2.938148405 Simplifying x + -1.722222222 = -2.938148405 Reorder the terms: -1.722222222 + x = -2.938148405 Solving -1.722222222 + x = -2.938148405 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.722222222' to each side of the equation. -1.722222222 + 1.722222222 + x = -2.938148405 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x = -2.938148405 + 1.722222222 x = -2.938148405 + 1.722222222 Combine like terms: -2.938148405 + 1.722222222 = -1.215926183 x = -1.215926183 Simplifying x = -1.215926183

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.660370627, -1.215926183}

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