24x^2-106x+81=0

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Solution for 24x^2-106x+81=0 equation:


Simplifying
24x2 + -106x + 81 = 0

Reorder the terms:
81 + -106x + 24x2 = 0

Solving
81 + -106x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
3.375 + -4.416666667x + x2 = 0

Move the constant term to the right:

Add '-3.375' to each side of the equation.
3.375 + -4.416666667x + -3.375 + x2 = 0 + -3.375

Reorder the terms:
3.375 + -3.375 + -4.416666667x + x2 = 0 + -3.375

Combine like terms: 3.375 + -3.375 = 0.000
0.000 + -4.416666667x + x2 = 0 + -3.375
-4.416666667x + x2 = 0 + -3.375

Combine like terms: 0 + -3.375 = -3.375
-4.416666667x + x2 = -3.375

The x term is -4.416666667x.  Take half its coefficient (-2.208333334).
Square it (4.876736114) and add it to both sides.

Add '4.876736114' to each side of the equation.
-4.416666667x + 4.876736114 + x2 = -3.375 + 4.876736114

Reorder the terms:
4.876736114 + -4.416666667x + x2 = -3.375 + 4.876736114

Combine like terms: -3.375 + 4.876736114 = 1.501736114
4.876736114 + -4.416666667x + x2 = 1.501736114

Factor a perfect square on the left side:
(x + -2.208333334)(x + -2.208333334) = 1.501736114

Calculate the square root of the right side: 1.225453432

Break this problem into two subproblems by setting 
(x + -2.208333334) equal to 1.225453432 and -1.225453432.

Subproblem 1

x + -2.208333334 = 1.225453432 Simplifying x + -2.208333334 = 1.225453432 Reorder the terms: -2.208333334 + x = 1.225453432 Solving -2.208333334 + x = 1.225453432 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.208333334' to each side of the equation. -2.208333334 + 2.208333334 + x = 1.225453432 + 2.208333334 Combine like terms: -2.208333334 + 2.208333334 = 0.000000000 0.000000000 + x = 1.225453432 + 2.208333334 x = 1.225453432 + 2.208333334 Combine like terms: 1.225453432 + 2.208333334 = 3.433786766 x = 3.433786766 Simplifying x = 3.433786766

Subproblem 2

x + -2.208333334 = -1.225453432 Simplifying x + -2.208333334 = -1.225453432 Reorder the terms: -2.208333334 + x = -1.225453432 Solving -2.208333334 + x = -1.225453432 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.208333334' to each side of the equation. -2.208333334 + 2.208333334 + x = -1.225453432 + 2.208333334 Combine like terms: -2.208333334 + 2.208333334 = 0.000000000 0.000000000 + x = -1.225453432 + 2.208333334 x = -1.225453432 + 2.208333334 Combine like terms: -1.225453432 + 2.208333334 = 0.982879902 x = 0.982879902 Simplifying x = 0.982879902

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.433786766, 0.982879902}

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