9x^2-31x+24=0

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Solution for 9x^2-31x+24=0 equation:


Simplifying
9x2 + -31x + 24 = 0

Reorder the terms:
24 + -31x + 9x2 = 0

Solving
24 + -31x + 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'.
2.666666667 + -3.444444444x + x2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + -3.444444444x + -2.666666667 + x2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + -3.444444444x + x2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + -3.444444444x + x2 = 0 + -2.666666667
-3.444444444x + x2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
-3.444444444x + x2 = -2.666666667

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 = -2.666666667 + 2.966049382

Reorder the terms:
2.966049382 + -3.444444444x + x2 = -2.666666667 + 2.966049382

Combine like terms: -2.666666667 + 2.966049382 = 0.299382715
2.966049382 + -3.444444444x + x2 = 0.299382715

Factor a perfect square on the left side:
(x + -1.722222222)(x + -1.722222222) = 0.299382715

Calculate the square root of the right side: 0.547158766

Break this problem into two subproblems by setting 
(x + -1.722222222) equal to 0.547158766 and -0.547158766.

Subproblem 1

x + -1.722222222 = 0.547158766 Simplifying x + -1.722222222 = 0.547158766 Reorder the terms: -1.722222222 + x = 0.547158766 Solving -1.722222222 + x = 0.547158766 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 = 0.547158766 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x = 0.547158766 + 1.722222222 x = 0.547158766 + 1.722222222 Combine like terms: 0.547158766 + 1.722222222 = 2.269380988 x = 2.269380988 Simplifying x = 2.269380988

Subproblem 2

x + -1.722222222 = -0.547158766 Simplifying x + -1.722222222 = -0.547158766 Reorder the terms: -1.722222222 + x = -0.547158766 Solving -1.722222222 + x = -0.547158766 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 = -0.547158766 + 1.722222222 Combine like terms: -1.722222222 + 1.722222222 = 0.000000000 0.000000000 + x = -0.547158766 + 1.722222222 x = -0.547158766 + 1.722222222 Combine like terms: -0.547158766 + 1.722222222 = 1.175063456 x = 1.175063456 Simplifying x = 1.175063456

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.269380988, 1.175063456}

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