3x^2-14x+9=0

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Solution for 3x^2-14x+9=0 equation:


Simplifying
3x2 + -14x + 9 = 0

Reorder the terms:
9 + -14x + 3x2 = 0

Solving
9 + -14x + 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'.
3 + -4.666666667x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.666666667x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.666666667x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.666666667x + x2 = 0 + -3
-4.666666667x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.666666667x + x2 = -3

The x term is -4.666666667x.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667x + 5.444444448 + x2 = -3 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667x + x2 = -3 + 5.444444448

Combine like terms: -3 + 5.444444448 = 2.444444448
5.444444448 + -4.666666667x + x2 = 2.444444448

Factor a perfect square on the left side:
(x + -2.333333334)(x + -2.333333334) = 2.444444448

Calculate the square root of the right side: 1.563471921

Break this problem into two subproblems by setting 
(x + -2.333333334) equal to 1.563471921 and -1.563471921.

Subproblem 1

x + -2.333333334 = 1.563471921 Simplifying x + -2.333333334 = 1.563471921 Reorder the terms: -2.333333334 + x = 1.563471921 Solving -2.333333334 + x = 1.563471921 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + x = 1.563471921 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + x = 1.563471921 + 2.333333334 x = 1.563471921 + 2.333333334 Combine like terms: 1.563471921 + 2.333333334 = 3.896805255 x = 3.896805255 Simplifying x = 3.896805255

Subproblem 2

x + -2.333333334 = -1.563471921 Simplifying x + -2.333333334 = -1.563471921 Reorder the terms: -2.333333334 + x = -1.563471921 Solving -2.333333334 + x = -1.563471921 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + x = -1.563471921 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + x = -1.563471921 + 2.333333334 x = -1.563471921 + 2.333333334 Combine like terms: -1.563471921 + 2.333333334 = 0.769861413 x = 0.769861413 Simplifying x = 0.769861413

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.896805255, 0.769861413}

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